paper a lower bound for generalized hamming weights ...the generalized hamming weights, which is...

1090IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.6 JUNE 1999

PAPER

A Lower Bound for Generalized Hamming Weights

and a Condition for t-th Rank MDS∗

Tomoharu SHIBUYA†, Member, Ryo HASEGAWA†, Nonmember,and Kohichi SAKANIWA†, Member

SUMMARY In this paper, we introduce a lower bound forthe generalized Hamming weights, which is applicable to arbi-trary linear code, in terms of the notion of well-behaving. Wealso show that any [n, k] linear code C over a finite field F isthe t-th rank MDS for t such that g(C) + 1 ≤ t ≤ k where g(C)is easily calculated from the basis of Fn so chosen that whosefirst n − k elements generate C⊥. Finally, we apply our resultto Reed-Solomon, Reed-Muller and algebraic geometry codes onCab, and determine g(C) for each code.key words: generalized Hamming weights, t-th rank MDS, Reed-Solomon codes, Reed-Muller codes, codes on affine algebraic va-riety, AG codes on Cab

1. Introduction

The notion of generalized Hamming weights was firstintroduced in [1]. Since then, lots of authors have inves-tigated generalized Hamming weights and have derivedsome estimates or true weights for several codes.

Recently in [2], a bound of generalized Hammingweights, called the order bound , was introduced in ageneral setting of codes on algebraic varieties which in-cludes one-point algebraic geometry codes as well asq-ary Reed-Muller codes. Especially, it was shown thatthe order bound agrees with the true generalized Ham-ming weight for Reed-Muller codes.

The order bound is based on the order function onan F -algebra (F : finite field) [3], which originates fromvaluation theory [4] and the theory of Grobner bases [5].It is known that when codes with code length n over Fare regarded as images of some linear evaluation mapfrom F -algebra into Fn [2], [6], [7], the order function isa useful tool to represent the well-behaving propertieson F -algebra, which was introduced to define Feng-Raodesigned distance [8].

In [7], [9], another lower bound for the generalizedHamming weights of codes constructed on affine alge-braic varieties was introduced by using the notion ofwell-behaving and some structures on the codes, suchas Grobner bases of ideals of affine algebraic varietiesand monomial orders.

Manuscript received July 10, 1998.Manuscript revised December 1, 1998.

†The authors are with the Department of Electricaland Electronic Engineering, Tokyo Institute of Technology,Tokyo, 152–8552 Japan.

∗This research is partially supported by the Ministry ofEducation, Japan, Grant No. 10750265.

In this paper, as a generalization of [7], [9], we firstintroduce a lower bound for the generalized Hammingweights of arbitrary linear code in terms of the notionof well-behaving. In this paper we only assume that weare given a sequence of vectors, B := {h1,h2, . . . ,hn},which is a basis of Fn and whose first n − k elementsconstitute the row vectors of a parity check matrix ofthe [n, k] code C.

Next, we introduce a parameter gB(C), which isuniquely determined from the basis B, and show thatthe t-th generalized Hamming weight of [n, k] linearcode C is equal to n− k + t for gB(C) + 1 ≤ t ≤ k. Acode whose t-th generalized Hamming weight is equalto n − k + t is said to be t-th rank MDS (MaximumDistance Separable) [1]. Thus we can say that any lin-ear code C is the t-th rank MDS for gB(C)+ 1 ≤ t ≤ kwhich, compared to the conventional sufficient condi-tions [10], gives a new type of sufficient condition forthe t-th rank MDS codes. Moreover, we note that com-puting gB(C) from the given basis B is a rather easytask.

Finally, we apply our result to some well-knowncodes, i.e., Reed-Solomon (RS) codes, Reed-Muller(RM) codes and algebraic geometry (AG) codes on thecurve called Cab. Then we show that gB(C) for RS andRM codes can be determined explicitly and the rangeof t for which these codes are the t-th rank MDS isthe same as the conventional result. As for AG codeson Cab, we show that gB(C) is upper bounded by thegenus of the curve Cab and the range of t for which thecode is the t-th rank MDS is wider than the conven-tional result.

2. Preliminaries

Let F be a finite field and for a subset A of Fn, wedenote by Supp (A) the support of A, i.e.,

Supp (A) := {i : 1 ≤ i ≤ n, ci �= 0for some c = (c1, c2, . . . , cn) ∈ A}.

Definition 1 [1] Let C be an [n, k] linear code over F .We denote by Dt the set of all t-dimensional subcodesof C for 1 ≤ t ≤ k. Then the t-th generalized Hammingweight of C is defined by

dt(C) := min{|Supp (D)| : D ∈ Dt}, (1)

SHIBUYA et al: A LOWER BOUND FOR GENERALIZED HAMMING WEIGHTS AND A CONDITION FOR T -TH RANK MDS1091

where |S| denotes the cardinality of a set S. ✷

The following results for the generalized Hammingweights are well known.

Proposition 1 [1], [11] For any linear [n, k] code Cover F , we have:

(i) Monotonicity:

1 ≤ d1(C) < d2(C) < · · · < dk(C) ≤ n.

(ii) Generalized Singleton bound:

dt(C) ≤ n− k + t for all t, 1 ≤ t ≤ k.

(iii) Duality: Let C⊥ be the dual code of C. Then

{dt(C)}kt=1 ∪ {n+ 1− dt(C⊥)}n−k

t=1

= {1, 2, . . . , n}.✷

Hereafter, let B := {h1,h2, . . . ,hn} denote a basisof Fn. It is noted that when we say that B is a basisof Fn, we include the order of hi’s in B.

We denote by Li := 〈h1,h2, . . . ,hi〉 (1 ≤ i ≤ n)the linear space over F spanned by {h1,h2, . . . ,hi},the first i elements of B, and let L0 := {0}.

For u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn)in Fn, u · v denotes inner product of u and v, that is,u · v :=

∑ni=1 uivi.

Definition 2 For {h1,h2, . . . ,hr} ⊂ B (1 ≤ r ≤ n),we define [n, n− r] code Cr by

Cr := L⊥r = 〈h1,h2, . . . ,hr〉⊥

= {c ∈ Fn : c · hi = 0 for all i = 1, 2, . . . , r}.✷

Definition 2 means that h1,h2, . . . ,hr are the rowvectors of a parity check matrix of Cr.

Definition 3 For given D ∈ Dt, we define an F -linear map θD : Fn → Fn, v = (v1, v2, . . . , vn) �→ vD =(vD

1 , vD2 , . . . , vD

n ) by

vD

i :={

vi, if i ∈ Supp (D),0, if i /∈ Supp (D).

✷

For D ∈ Dt, we denote by SDi := 〈hD

1,hD

2, . . . ,hD

i 〉(1 ≤ i ≤ n) the linear space over F spanned by{hD

1,hD

2, . . . ,hD

i } and let SD0 := {0}. In general, vectors

hD

1,hD

2, . . . ,hD

n are not necessarily linearly independentand dimSD

i ≤ i.

Proposition 2 For any D ∈ Dt, |Supp (D)| =dimSD

n .(Proof) It is obvious that

Ker(θD)= {v = (v1, v2, . . . , vn) ∈ Fn :

vi = 0 for all i ∈ Supp (D)}.This implies that dim(Ker(θD)) = n−|Supp (D)|. Since{h1,h2, . . . ,hn} is a basis of Fn, we see SD

n = Im(θD).Thus we have

dimSD

n =dim(Im(θD))=dim(Fn)− dim(Ker(θD))= |Supp (D)|.

✷

For u = (u1, u2, . . . , un),v = (v1, v2, . . . , vn) ∈ Fn,we denote u ∗ v := (u1v1, u2v2, . . . , unvn) ∈ Fn.

Definition 4 [12], [13] We define the map ρ : Fn →{0, 1, 2, . . . , n} by

ρ(v) :={0, if v = 0,k, if v(�= 0) ∈ Lk \ Lk−1.

A pair (hi,hj) (hi,hj ∈ B, 1 ≤ i, j ≤ n) is said to bewell-behaving if ρ(hu ∗hv) < ρ(hi ∗hj) for all hu,hv ∈B with 1 ≤ u ≤ i, 1 ≤ v ≤ j and u+ v < i+ j. ✷

For each hi ∈ B, we define

Λi := {k : k ∈ {0, 1, 2, . . . , n} such thatk = ρ(hi ∗ hj) wherehj ∈ B and (hi,hj) is well-behaving}.

(2)

For a subset T of {1, 2, . . . , r}, we also define

ΛT :=⋃i∈T

Λi,

Λ∗T := {r + 1, r + 2, . . . , n} \ ΛT .

(3)

For given D ∈ Dt, let

TD := {i : 1 ≤ i ≤ r,hD

i ∈ SD

i−1}.

Remark 1 It can be seen from Definition 4 that ρdepends on B and the order of hi’s in B. Thus so doΛi, ΛT and Λ∗

T . ✷

3. A Lower Bound for Generalized HammingWeights and Condition for t-th Rank MDS

3.1 Main Results

Let Dt be the set of all t-dimensional subcodes of Cr

with 1 ≤ t ≤ n− r. For D ∈ Dt and a linear subspaceW ⊂ Fn, we define W⊥D by

W⊥D := {v ∈ Fn : Supp (v) ⊂ Supp (D)and v · u = 0 for all u ∈ W}.

Since D ⊂ Cr, for all c = (c1, c2, . . . , cn) ∈ D andhi = (hi1, hi2, . . . , hin) (1 ≤ i ≤ r), we have


0 = c · hi =n∑

j=1

cjhij =∑

j∈Supp(D)

cjhij = c · hD

i ,

which means

D ⊂ 〈hD

1,hD

2, . . . ,hD

r〉⊥D = (SD

r )⊥D . (4)

In order to derive a lower bound of generalizedHamming weights, we need a couple of lemmas.

Lemma 1 For any D ∈ Dt, there exist at least telements hD

i ’s in {hD

r+1,hD

r+2, . . . ,hD

n} such that hD

i /∈SD

i−1, r + 1 ≤ i ≤ n.(Proof) Assume that there exist only µ (≤ t − 1) el-ements, denoted by hD

i1 ,hD

i2 , . . . ,hD

iµ, in {hD

r+1,hD

r+2,

. . . ,hD

n} which satisfy hD

ij/∈ SD

ij−1. Then we can write

SD

n =SD

r + 〈hD

r+1,hD

r+2, . . . ,hD

n〉=SD

r ⊕ 〈hD

i1 , . . . ,hD

iµ〉 (5)

where ⊕ denotes direct sum. Hence, by noting thatdimSD

n = dimSDr + dim(SD

r )⊥D , we have from Eqs.(4)and (5) that

dimD≤dim(SD

r )⊥D = dimSD

n − dimSD

r

=dim 〈hD

i1 , . . . ,hD

iµ〉 = µ ≤ t− 1

which contradicts with dimD = t. ✷

Lemma 2 Let (hi,hj) (hi,hj ∈ B) be well-behaving and k := ρ(hi ∗ hj). For given D ∈ Dt, ifhD

i ∈ SDi−1 or hD

j ∈ SDj−1, then hD

k ∈ SD

k−1.(Proof) Since k = ρ(hi ∗hj), hi ∗hj can be expressedas hi ∗ hj =

∑kν=1 ανhν with hν ∈ B, αν ∈ F and

αk �= 0. Thus by noting that hD

i ∗ hD

j = θD(hi ∗ hj),hD

i ∗ hD

j is expressed as

hD

i ∗ hD

j =k∑

ν=1

ανhD

ν , hν ∈ B,αν ∈ F , αk �= 0. (6)

Without loss of generality, we assume that hD

i ∈SD

i−1. Since hD

j ∈ SD

j , hD

i ∗ hD

j can be also expressed as

hD

i ∗ hD

j =

(i−1∑u=1

auhD

u

)∗(

j∑v=1

bvhD

v

)

=∑

1≤u≤i−1,1≤v≤j

βu,vhD

u ∗ hD

v

=∑

1≤u≤i−1,1≤v≤j

βu,vθD(hu ∗ hv),

hu,hv ∈ B, au, bv, βu,v ∈ F .

Since (hi,hj) is well-behaving, ρ(hu ∗hv) < k for every0 ≤ u ≤ i− 1 and 0 ≤ v ≤ j. Hence

hD

i ∗ hD

j =k−1∑ν=1

βνhD

ν , hν ∈ B, βν ∈ F . (7)

Therefore, we have from Eqs. (6) and (7) that

hD

k =1αk

k−1∑ν=1

(βν − αν)hν

which implies that hD

k ∈ SD

k−1. ✷

Lemma 3 For any hk ∈ B (r + 1 ≤ k ≤ n) andgiven D ∈ Dt, if hD

k /∈ SD

k−1, then k /∈ ΛTD .(Proof) We show the contraposition. For any k ∈ΛTD = ∪i∈TDΛi, there exists some i ∈ TD such thatk ∈ Λi. Therefore, by the definition of Λi, there existssome hj ∈ B such that k = ρ(hi ∗ hj) and (hi,hj) iswell behaving.

On the other hand, by the definition of TD, hD

i ∈SD

i−1 for any i ∈ TD. Thus by Lemma 2 we have hD

k ∈SD

k−1. ✷

Theorem 1 For [n, n − r] code Cr given in Defini-tion 2, let

ηt := r −max{|T | : T ⊂ {1, 2, . . . , r}such that |Λ∗

T | ≥ t}. (8)

Then dt(Cr) ≥ ηt + t for any t, 1 ≤ t ≤ n− r.(Proof) Note that for D ∈ Dt, we have defined TD :={i : 1 ≤ i ≤ r, hD

i ∈ SDi−1}, which yields

|{i : 1 ≤ i ≤ r,hD

i /∈ SD

i−1}| = r − |TD|. (9)

On the other hand, we have from Lemma 1 that

|{i : r + 1 ≤ i ≤ n,hD

i /∈ SD

i−1}| ≥ tfor any D ∈ Dt.

}(10)

Thus if r − |TD| ≥ ηt for any D ∈ Dt, we have fromProposition 2, Eqs.(9) and (10) that

|Supp (D)|=dim 〈hD

1, . . . ,hD

n〉= |{i : 1 ≤ i ≤ n,hD

i /∈ SD

i−1}|= |{i : 1 ≤ i ≤ r,hD

i /∈ SD

i−1}|+|{i : r + 1 ≤ i ≤ n,hD

i /∈ SD

i−1}|≥ ηt + t.

Therefore it is sufficient to show that r − |TD| ≥ ηt forany D ∈ Dt.

For each i (r + 1 ≤ i ≤ n), if hD

i /∈ SDi−1 then

i /∈ ΛTD by Lemma 3, which means

{i : r + 1 ≤ i ≤ n,hD

i /∈ SD

i−1}⊂ {r + 1, r + 2, . . . , n} \ ΛTD = Λ∗

TD. (11)

Thus we see from Eqs.(10) and (11) that |Λ∗TD

| ≥ tfor any D ∈ Dt. Moreover, by noting that TD ⊂{1, 2, . . . , r}, we have

{TD : D ∈ Dt} ⊂ {T ⊂ {1, 2, . . . , r} : |Λ∗T | ≥ t}.

Therefore for any D ∈ Dt


r − |TD|≥ r −max{|TD′ | : D′ ∈ Dt}≥ r −max{|T | : T ⊂ {1, 2, . . . , r}, |Λ∗

T | ≥ t} =: ηt,

which completes the proof. ✷

By applying Proposition 1-(i) to this theorem, weimmediately obtain a slightly improved bound.

Corollary 1 For Cr and ηt, let η1 := η1 + 1 and

ηt := max{ηt + t, ηt−1 + 1}, t = 2, 3, . . . , n− r.

Then dt(Cr) ≥ ηt for 1 ≤ t ≤ n− r. ✷

Theorem 1 may look like a paraphrase of the orig-inal problem into an equally difficult question becausewe have to take all subsets T ’s in {1, 2, . . . , r} to calcu-late Eq. (8). However, as shown in the next theorem,we can obtain a further information on the generalizedHamming weights of Cr via Theorem 1.

Theorem 2 For Cr, letAi := {r+1, r+2, . . . , n}\Λi

(i = 1, 2, . . . , r) and gB(Cr) := max{|Ai| : 1 ≤ i ≤ r}.Then dt(Cr) = r + t for all t, gB(Cr) + 1 ≤ t ≤ n− r.(Proof) Let T be a subset of {1, 2, . . . , r}. Since Λ∗

T ⊂Ai for all i ∈ T , |Λ∗

T | ≤ gB(Cr) for any T . So there isno T (�= ∅) ⊂ {1, 2, . . . , r} such that |Λ∗

T | ≥ gB(Cr) + 1.Thus, for t ≥ gB(Cr) + 1, ηt = r in Eq.(8) and we havedt(Cr) ≥ r + t.

On the other hand, dt(Cr) ≤ r + t byProposition 1-(ii). ✷

Remark 2 For given B and r, it is easy to calculateΛi (1 ≤ i ≤ r) by using, for example, Gaussian elimi-nation. Thus it is relatively easy to obtain gB(Cr) inTheorem 2. By the same reason as mentioned in Re-mark 1, gB(Cr) also depends on B and the order of hi’sin B. ✷

An [n, n − r] code C is called the t-th rank MDSif an equality holds in the generalized Singleton bound,that is, dt(C) = r+ t [1], and some sufficient conditionsfor the t-th rank MDS are discussed in [10]. Since any[n, n− r] linear code Cr can be expressed as in Defini-tion 2, we see from Theorem 2 that Cr is the t-th rankMDS for gB(Cr)+1 ≤ t ≤ n−r, which gives a new typeof sufficient condition for the t-th rank MDS codes.

3.2 Lower Bound for Generalized Hamming Weightsof Dual Codes

Here we investigate generalized Hamming weights ofC⊥

r .

Theorem 3 Define {δ1, δ2, . . . , δr} (δi < δi+1) by

{δ1, δ2, . . . , δr} := {1, 2, . . . , n} \ {n+ 1− ην}n−rν=1 .

Then dt(C⊥r ) ≥ δt for 1 ≤ t ≤ r.

(Proof) We have from Corollary 1 that

n+ 1− dν(Cr) ≤ n+ 1− ην (12)

for 1 ≤ ν ≤ n − r and in particular with equality forgB(Cr)+1 ≤ ν ≤ n−r by Theorem 2. By Proposition 1-(i) and (iii), dt(C⊥

r ) is the t-th smallest element in

{1, 2, . . . , n} \ {n+ 1− dν(Cr)}n−rν=1

and δt is the t-th smallest element in {δ1, δ2, . . . , δr}.Therefore we have from Eq.(12) that dt(C⊥

r ) ≥ δt (1 ≤t ≤ r). ✷

Corollary 2 Let

δt :=

n− r − gB(Cr) + 1, for t = 1,δt, for 2 ≤ t ≤ r − η1 + 1,n− r + t, for r − η1 + 2 ≤ t ≤ r.

Then dt(C⊥r ) ≥ δt with equality for r − η1 + 2 ≤ t ≤ r.

(Proof) (i) In the case t = 1: By Corollary 1 andTheorem 2, we can write

{n+ 1− dν(Cr)}n−rν=1

= {1, 2, . . . , n− r − gB(Cr),n+ 1− dgB(Cr)(Cr), . . . , n+ 1− d1(Cr)}.

Hence by Proposition 1-(iii) d1(C⊥r ) is not less than

n− r − gB(Cr) + 1.(ii) In the case 2 ≤ t ≤ r − η1 + 1: Trivial from

Theorem 3.(iii) In the case r−η1+2 ≤ t ≤ r: We have max{n+

1 − dν(Cr)}n−rν=1 = n + 1 − d1(Cr) by Proposition 1-(i)

and n+ 1− d1(Cr) ≤ n− η1 + 1 by Corollary 1. Thuswe have from Proposition 1-(iii) that all integers i suchthat n − η1 + 2 ≤ i ≤ n are not included in {n + 1 −dν(Cr)}n−r

ν=1 , that is, {n − η1 + 2, . . . , n} are includedin {dt(C⊥

r )}rt=1 and are the largest η1 − 1 elements in

{1, 2, . . . , n}. Therefore, by Proposition 1-(i), we have

{n− η1 + 2, . . . , n} = {dt(C⊥r )}r

t=r−η1+2

and dt(C⊥r ) = n− r + t. ✷

3.3 Comparison of the Proposed Bound with the Or-der Bound

In this subsection, we compare the proposed boundwith the order bound [2] from some technical points ofview.

(a) A major advantage of the proposed boundwould be an introduction of gB(Cr) which, as shownin the next section, gives a rather fine range of t forwhich RS, RM and AG codes are t-th rank MDS.

(b) It is seen that the class of codes to which theproposed bound can be applied is wider than that forthe order bound.


We can see that the order bound can not be ap-plied to all linear codes since it requires an order func-tion which must satisfy some specific conditions andtherefore can not always produce a basis of Fn whichincludes the vectors of a parity check matrix of a givenlinear code as its first r elements. It is also noted thatno concrete procedure to construct an order functionfor a given code is presented in [2].

On the other hand, the proposed bound can becomputed for any linear code because a basis B ={h1,h2, . . . ,hn} of Fn can be obtained by simplyadding independent vectors to the vectors of a paritycheck matrix of a given linear code.

(c) As for the tightness of the bounds, as we showsome numerical examples in the next section, the pro-posed bound has given no worse value so far than theconventional ones including the order bound.

However, it seems difficult to make a general com-parison between the two bounds and we must leave itfor further study to clarify the tightness of the bounds.

(d) Finally, we shall roughly compare the compu-tational complexity to calculate these two bounds.

As is seen from Eq. (8), the proposed bound needsto verify whether |Λ∗

T | ≥ t for all subsets T ’s of{1, 2, . . . , r} and find the maximum value of |T | with|Λ∗

T | ≥ t. The complexity to verify if |Λ∗T | ≥ t for

T is proportional to |T |. Since the number of subsetsT ’s of {1, 2, . . . , r} with |T | = i is

(ri

), the complexity

is proportional to∑r

i=1 i(ri

)which increases as r in-

creases and does not depend on t. It is noted that fort ≥ gB(Cr) + 1, the proposed bound requires no calcu-lation as Theorem 2 gives the exact value dt(Cr) = r+tfor the generalized Hamming weight.

On the other hand, the order bound needs to eval-uate a function, denoted by a(%1, %2, . . . , %t) in [2], for allt-tuples (%1, %2, . . . , %t) such that %i ∈ {r+1, r+2, . . . , n}and %1 < %2 < · · · < %t, and to find the minimum valueof a(%1, %2, . . . , %t). Provided that an order function isgiven, the complexity to calculate a(%1, %2, . . . , %t) for at-tuple (%1, %2, . . . , %t) is proportional to t. Since thereare

(n−r

t

)such t-tuples, the complexity is proportional

to t(n−r

t

), which increases as r decreases, and increases

with t for 1 ≤ t ≤ (n − r + 1)/2 and decreases with tfor (n− r + 1)/2 ≤ t ≤ n− r.

As an example,∑r

i=1 i(ri

)and t

(n−r

t

)are compared

for n = 64 in Figs. 1 and 2. We can see from Fig. 1that the computational complexity for the proposedbound is smaller than or comparable to that for theorder bound when the rate of a code is relatively large(i.e., for a smaller r), and the complexities for the twobounds are complementary with respect to r.

In Fig. 2, we compare the complexities of the twobounds for r = 8 and 16. Figure 2 shows that thecomplexity for the proposed bound is much less thanthat for the order bound in most of t.

It is noted again that while the complexity for the

Fig. 1 Comparison of complexity (1).

Fig. 2 Comparison of complexity (2).

proposed bound is drawn for whole range of r and t inFigs. 1 and 2, respectively, no computation is requiredfor t ≥ gB(Cr) + 1.

4. Applications

In this section, we apply Theorems 1 and 2 to a cou-ple of representative codes, i.e., Reed-Solomon codes,Reed-Muller codes and codes constructed on affine al-gebraic varieties which contain conventional one-pointalgebraic geometry codes. Then we show that gB(Cr)for RS and RM codes can be determined explicitly andgB(Cr) for AG codes on Cab is upper bounded by thegenus g of the curve Cab.

4.1 Reed-Solomon Codes

Let α be a primitive element of F := GF (q) and n :=q − 1. We set the basis of Fn as B = {h1,h2, . . . ,hn}where

hi := (1, αi−1, α2(i−1), . . . , α(n−1)(i−1)) ∈ Fn.

Then Cr becomes the [n, n − r, r + 1] Reed-Solomon(RS) code.


For each hi (1 ≤ i ≤ r),

hi ∗ hj =(1, αi+j−2, α2(i+j−2), . . . , α(n−1)(i+j−2))=hi+j−1, j = 1, 2, . . . , n− i+ 1,

which implies that ρ(hi ∗ hj) = i + j − 1. Moreover,it can be easily verified that for each hi (1 ≤ i ≤ r),(hi,hj) are all well-behaving for j = 1, 2, . . . , n− i+1.Thus we have

Λi = {i, i+ 1, . . . , n}and

Ai = {r + 1, r + 2, . . . , n} \ Λi = ∅for all i = 1, 2, . . . , r. Therefore, by the definition, wehave gB(Cr) = 0.

By Theorem 2, it holds that dt(Cr) = r + t for allt (1 ≤ t ≤ n− r), which implies that RS codes are thet-th rank MDS codes for 1 ≤ t ≤ n − r. This result isa well known fact [1], [10].

4.2 Reed-Muller Codes

Let R be the polynomial ring over F := GF (q) withm variables, i.e., R := F [X1, X2, . . . , Xm]. We alsolet P be the set of all distinct points of Fm, that is,P = {P1, P2, . . . , Pn} where n = qm. For f ∈ R andP , let ψ(f) := (f(P1), f(P2), . . . , f(Pn)). The map ψ :R → Fn is a surjective homomorphism of F -algebra [2].

We define deg(f) :=∑m

�=1 i� for a monomial f =∏m�=1 X i�

� ∈ R and deg(f) := max{deg(fi)} for f =∑i fi where fi ∈ R denote monomials.

Definition 5 [2] The q-ary Reed-Muller code of orderu and in m variables is defined by

RMq(u,m) := {ψ(f) : f ∈ R, deg(f) ≤ u}.✷

A monomial∏m

�=1 X i�

� ∈ R is said to be reduced if0 ≤ i� ≤ q − 1 for all % (1 ≤ % ≤ m). There are qm

(= n) reduced monomials in R, and it is shown in [2]that

RMq(u,m)= 〈{ψ(f) : f ∈ R is a reducedmonomial with deg(f) ≤ u}〉.

(13)

In [2], graded lexicographic order , which is one of themonomial orders on R [5], is employed to constructfrom {ψ(f) : f ∈ R is a reduced monomial} a basisof Reed-Muller code.

Definition 6 (Graded lexicographic order ≺GL)[5] For fi =

∏m�=1 X i�

� and fj =∏m

�=1 Xj�

� in R, wesay fi ≺GL fj if (i) deg(fi) < deg(fj), or (ii) deg(fi) =deg(fj) and fi ≺L fj , where ≺L denotes a lexicographicorder [5], i.e., fi ≺L fj if, in the vector (j1 − i1, j2 −i2, . . . , jm−im), there exists nonzero entry and the left-most nonzero entry is positive. ✷

Write

{f ∈ R : f is a reduced monomial}= {f1, f2, . . . , fn} =: Γq

with fi ≺GL fi+1 and n = qm, and define

B := {h1,h2, . . . ,hn}, hi := ψ(fi).

We also let Γq(u) := {f ∈ Γq : deg(f) ≤ u}. Then it isobvious from Eq.(13) that

RMq(u,m)= span{ψ(f) : f ∈ Γq(u)}= 〈h1,h2, . . . ,hk〉

where k := |Γq(u)|.Proposition 3 [2] The dual code of RMq(u,m) isRMq(m(q − 1)− u− 1, m). ✷

By this proposition, we see that

RMq(u,m) = 〈h1,h2, . . . ,hr〉⊥ = Cr

where r := |Γq(m(q − 1) − u − 1)|, and have our finalresult as follows.

Theorem 4 Consider Cr = RMq(u,m) where r =|Γq(m(q − 1) − u − 1)|. Let Q and R be integers suchthat

m(q − 1)− u− 1 = Q(q − 1) +R,0 ≤ Q, 0 ≤ R ≤ q − 2.

}(14)

Then gB(Cr) = qm − r − (q −R)qm−(Q+1) + 1. (Proofis given in the Appendix.) ✷

A method to compute t-th generalized Hammingweights for RMq(u,m) has been shown in [1] for q = 2and in [2] for arbitrary q. But no explicit condition on tfor RMq(u,m) to be t-th rank MDS has been given yetin terms of parameters of RM codes such as q, u and m.On the other hand, we can get from Theorems 2 and 4an explicit condition: RMq(u,m) is t-th rank MDS fort satisfying

qm − r − (q −R)qm−(Q+1) + 2 ≤ t ≤ n− r

where r = |Γq(m(q − 1) − u − 1)| and Q and R are asgiven in Eq. (14).

Numerical Example 1 Here we consider RM3(2, 3)and RM3(3, 3). As described above, we have n = 33 =27, RM3(2, 3) = C17 and RM3(3, 3) = C10.

For C17, Q and R in Eq.(14) are Q = R = 1 andgB(C17) = 5. Hence dt(C17) = 17 + t for 6 ≤ t ≤ 10 byTheorem 2.

For C10, Q and R in Eq.(14) are Q = 1 and R = 0and therefore gB(C10) = 9. Thus dt(C10) = 10 + t for10 ≤ t ≤ 17 by Theorem 2.

For these two codes, we see from Examples 5.13and 5.14 in [2] that gB(C17) and gB(C10) give the max-imum range of t for which dt(Cr) = r + t holds. ✷


Table 1 The values for gB(Cr) of Hermitian code on x5 +y4 +y = 0 over GF (24) withorder ≺ab in B.

r 1 2, 3 4, 5, 6 7, . . . , 47 48 49, 50, 51 52 53 54, 55 56 57 58, 59 60 61, 62 63gB(Cr) 0 3 5 6 5 6 5 4 6 5 4 3 2 1 0

4.3 Codes on Affine Algebraic Varieties

The results in the previous subsection can be general-ized to codes on affine algebraic varieties.

Definition 7 [6], [13], [14] For a subset V ⊂ Fm, wedenote by I(V ) the ideal of the polynomial ring R :=F [X1, X2, . . . , Xm] given by

I(V ) := {f ∈ R : f(P ) = 0 for all P ∈ V }.We also denote by R(V ) the coordinate ring of V de-fined by R(V ) := R/I(V ). ✷

As in Sect. 4.2, for f ∈ R and V = {P1, P2, . . . , Pn}where n := |V |, let ψ(f) := (f(P1), f(P2), . . . , f(Pn)).Then it is shown in [6], [13], [14] that the coordinate ringR(V ) and Fn are isomorphic as linear spaces over Fand an isomorphism is given by the F -linear map ψ :R(V ) → Fn which is induced by ψ.

For a basis {f1, f2, . . . , fn} of R(V ) over F , letB = {h1,h2, . . . ,hn}, hi := ψ(fi). Then B becomesa basis of Fn over F and code Cr can be defined asin Definition 2 and is called a code constructed on anaffine algebraic variety V [6], [13], [14].

As an example, we consider the case in which V isthe set of rational points on the curve called Cab [15].It is known that the genus g of Cab is g = (a − 1)(b −1)/2 [15]. Let R := F [x, y] and

V := {(x, y) ∈ F 2 : h(x, y) = 0}where h(x, y) is the defining polynomial of Cab.

Definition 8 For positive integers a and b andmonomials xi1yj1 , xi2yj2 ∈ R, we say that xi1yj1 ≺ab

xi2yj2 if (i) ai1 + bj1 < ai2 + bj2, or (ii) ai1 + bj1 =ai2 + bj2 and i1 > i2. ✷

It is shown in [6], [13], [14] that ≺ab is a monomial or-der on R and we can take n monomials {f1, f2, . . . , fn}in R as a basis of R(V ), where fi ≺ab fi+1 (i =1, 2, . . . , n − 1). Now, for these fi’s, let hi := ψ(fi),B := {h1,h2, . . . ,hn} and construct Cr as in Defini-tion 2. Then we have:

Theorem 5 [7], [16] Let g be the genus of the curveCab. Then gB(Cr) ≤ g (1 ≤ r ≤ n). ✷

Hereafter, notations on AG codes follow [17]. LetF (x, y)/F with h(x, y) = 0 be an algebraic functionfield of genus g over a finite constant field F . Let{P1, P2, . . . , Pn} be a set of places of degree one in

F (x, y)/F . Let G and D be divisors of F (x, y)/F suchthat D = P1 + P2 + · · · + Pn and G = mQ where Qis the common pole of x and y. Then one-point AGcode on Cab, denoted by CL(D,G), is defined, and it isshown that Cr = CL(D,G)⊥ where r is the dimensionof CL(D,G) [6].

For the generalized Hamming weights of AG codeCL(D,G), the following proposition is known.

Proposition 4 [10], [11], [18] If degG > 2g − 2, theAG code CL(D,G) has dt(CL(D,G)) = n−k+ t for allt such that g + 1 ≤ t ≤ k, where k is the dimension ofCL(D,G). ✷

It is also known that CL(D,G)⊥ = CL(D,H) forsome divisor H (see [17] for the detail). Thus we havethe translation of Proposition 4 for the dual code ofCL(D,G).

Corollary 3 Notations are as above. If degH >2g − 2, the dual of the AG code CL(D,G)⊥ (=CL(D,H)) has dt(CL(D,G)⊥) = k + t for all t suchthat g + 1 ≤ t ≤ n − k, where k is the dimension ofCL(D,G). ✷

By Corollary 3, we have dt(Cr) = r+ t for g+1 ≤t ≤ n − r where g is the genus of Cab. On the otherhand, we see from Theorems 2 and 5 that dt(Cr) = r+tfor gB(Cr) + 1 ≤ t ≤ n − r with gB(Cr) ≤ g. Thuswe can conclude that gB(Cr) gives the range of t notnarrower than that given in [10], [11], [18], for which thegeneralized Singleton bound holds with equality.

Numerical Example 2 Let a = 4, b = 5 and con-sider the curve defined by h(x, y) = x5 + y4 + y overGF (24). This curve is known as a Hermitian curve andits genus is g = 6.

The values of gB(Cr) obtained from its defini-tion given in Theorem 2 are listed in Table 1. Wecan see from Table 1 that gB(Cr) < g for r =1, . . . , 6, 48, 52, 53, 56, . . . , 63. For these r, the range oft for which dt(Cr) = r+t holds is wider than that givenin [10], [11], [18]. ✷

In the discussion in [7], the order of elements in Bmust satisfy the conditions of monomial order, whileany order in B is accepted in the discussion in thispaper. Thus the class of codes to which the bound canbe applied is wider for the proposed bound comparedwith the bound provided in [7].

The following example shows that the order of el-ements of B led by a monomial order does not alwaysgive the best value of gB(Cr).


Numerical Example 3 Let α be a primitive ele-ment of F := GF (23) and consider a code on affinealgebraic variety

V := {(0, 0), (0, 1), (1, α), (1, α2),(1, α4), (α,α5), (α2, α3), (α4, α6)}

=: {P1, P2, . . . , P8}.We employ a monomial order on R defined by Defini-tion 8 with a = 3 and b = 4. Then a basis of R(V ) isgiven by†

{1, x, y, x2, xy, y2, x3, x2y} =: {f1, f2, . . . , f8}and we have

B = {h1,h2, . . . ,h8},hi := ψ(fi) = (fi(P1), fi(P2), . . . , fi(P8)).

The values of gB(Cr) obtained from the definition givenin Theorem 2 are

gB(C1) = 0, gB(C2) = · · · = gB(C6) = 2,gB(C7) = 1.

For any basis B′ = {h′1,h

′2, . . . ,h

′8} of F 8, which is

obtained by changing the order of elements of B, suchthat

〈h′1,h

′2, . . . ,h

′r〉⊥ = 〈h1,h2, . . . ,hr〉⊥ = Cr,

we can verify by computer search that gB(Cr) ≤gB′(Cr) for r = 1, 2, . . . , 5 and 7, while for r = 6, wecan find that B′’s given by

{h1,h3,h2,h6,h5,h4,h7,h8},{h1,h3,h5,h2,h6,h4,h7,h8},{h1,h6,h3,h4,h2,h5,h7,h8},{h1,h2,h3,h6,h4,h5,h7,h8}

yield gB′(C6) = 1 < gB(C6). Moreover, we can showthat these four B′’s are not obtained from a monomialorder considered in [7].

(a) We first show that {f1, f2, . . . , f8} is the onlyset of monomial in R with ψ(fi) = hi and fi = fi

G

(i = 1, 2, . . . , 8).Let Fi := {fi + h : h ∈ I(V )}. Then since I(V ) is

the kernel of ψ : R → F 8, it is obvious that ψ(f) = hi

if and only if f ∈ Fi.Choose next an arbitrary monomial f ′

i ∈ Fi (i =1, 2, . . . , 8) such that {f ′

1, f′2, . . . , f

′8} �= {f1, f2, . . . , f8}.

Then {f ′1, f

′2, . . . , f

′8} is also a set of monomials in R

with ψ(f ′i) = hi. We show that f ′

i = f ′i

G(i =

1, 2, . . . , 8) cannot hold for this {f ′1, f

′2, . . . , f

′8}.

Let fj �= f ′j . Then since f ′

j

G= fj

Gby Proposi-

tion 5 in [7] and fjG

= fj by the footnote on p.1097,

we have f ′j

G= fj �= f ′

j . Thus we can conclude that{f1, f2, . . . , f8} is the only set of monomials in R suchthat ψ(fi) = hi and fi = fi

G(i = 1, 2, . . . , 8).

(b) In order to verify if there exists a monomialorder which leads any of the above four bases, it issufficient by (a) above to examine whether the set{f1, f2, . . . , f8} with its order so changed as it giveseach of the above four bases satisfies the condition of amonomial order.

Note that for f, g ∈ R and h ∈ R \ {0}, any mono-mial order ≺M must satisfy that fh ≺M gh if f ≺M g.It is easy to see that none of the above four bases satisfythis condition. In fact, for the first basis, for example,we must have y ≺ x and x2y � x3 at the same time,which contradicts the condition for a monomial order.Thus we can conclude that the above four bases are notobtained from a monomial order treated in [7]. ✷

5. Conclusion

In this paper, we have introduced a lower bound forgeneralized Hamming weights of arbitrary linear codeand its dual in terms of the notion of well-behaving.The proposed bound can be obtained only from thebasis B of Fn whose first r elements constitute theparity check matrix of the code and requires no otherstructure of the code. We have also shown that any[n, k] linear code C is the t-th rank MDS for gB(C) +1 ≤ t ≤ k where gB(C) is uniquely determined fromB. Finally, we have applied our results to RS codes,RM codes and AG codes on Cab. Then we have givenexplicit formulae of gB(C) for RS and RM codes andshown that gB(C) ≤ g holds for AG codes on Cab whereg is the genus of Cab.

As noted in Remarks 1 and 2, the proposed bounddepends on the choice of a basis B of Fn including theorder of vectors in B. It is remained for further studyto clarify which choice of B makes the bound tighter.

References

[1] V.K. Wei, “Generalized Hamming weights for linear codes,”IEEE Trans. Inf. Theory, vol.IT-37, no.5, pp.1412–1418,Sept. 1991.

[2] P. Heijnen and R. Pellikaan, “Generalized Hammingweights of q-ary Reed-Muller codes,” IEEE Trans. Inf. The-ory, vol.IT-44, no.1, pp.181–196, Jan. 1998.

[3] R. Pellikaan, “On the existence of order functions,” to ap-pear in Journal of Statistical Planning and Inference, 1996.

[4] O. Endler, “Valuation Theory (Universitext),” Springer-Verlag, Berlin, Germany, 1972.

[5] D. Cox, J. Little, and D. O’Shea, “Ideals, varieties,and algorithms: An introduction to computational alge-braic geometry and commutative algebra, Second edition,”Springer-Verlag, 1996.

[6] S. Miura, “On error correcting codes based on algebraicgeometry,” Ph.D. dissertation, Univ. of Tokyo, 1997.

[7] T. Shibuya, J. Mizutani, and K. Sakaniwa, “On generalizedHamming weights of codes constructed on affine algebraic

†A basis of R(V ) obtained in [7] always consists of mono-

mials of the form fG

= f , where fG

denotes the remainderon division of f by Grobner basis G of I(V ).


varieties,” IEICE Trans. Fundamentals, vol.E81-A, no.10,pp.1979–1989, Oct. 1998.

[8] G.L. Feng and T.R.N. Rao, “A simple approach for constric-tion of algebraic-geometric codes from Affine plane curves,”IEEE Trans. Inf. Theory, vol.IT-40, no.4, pp.1003–1012,July 1994.

[9] T. Shibuya, J. Mizutani, and K. Sakaniwa, “On generalizedHamming weights of codes constructed on Affine algebraicsets,” Proc. AAECC-12, Lecture Notes in Computer Sci-ence, vol.1255, Springer-Verlag, pp.311–320, 1997.

[10] M.A. Tsfasman and S.G. Vladut, “Geometric approach tohigher weights,” IEEE Trans. Inf. Theory, vol.IT-41, no.6,pp.1564–1588, Nov. 1995.

[11] K.Y. Yang, P.V. Kumar, and H. Stichtenoth, “On theweight hierarchy of geometric Goppa codes,” IEEE Trans.Inf. Theory, vol.IT-40, no.3, pp.913–920, May 1994.

[12] G.L. Feng and T.R.N. Rao, “Improved geometric Goppacodes,” IEEE Trans. Inf. Theory, vol.IT-41, no.6, pp.1678–1693, Nov. 1995.

[13] S. Miura, “Geometric Goppa codes on Affine algebraic va-riety,” Proc. 18th SITA, pp.243–246, Nov. 1995

[14] S. Miura, “Linear codes on Affine algebraic varieties,”IEICE Trans., vol.J81-A, no.10, pp.1386–1397, Oct. 1998.

[15] S. Miura, “Algebraic geometric codes on certain planecurves,” IEICE Trans., vol.J75-A, no.11, pp.1735–1745,Nov. 1992.

[16] T. Shibuya, J. Mizutani, and K. Sakaniwa, “On lowerbound of Generalized Hamming weights for codes on Cab,”Proc. 20th SITA, vol.2, pp.853–856, Dec. 1997.

[17] H. Stichtenoth, “Algebraic Function Fields and Codes,”Springer-Verlag, 1993.

[18] C. Munuera, “On the generalized Hamming weights of ge-ometric Goppa codes,” IEEE Trans. Inf. Theory, vol.IT-40,no.6, pp.2092–2099, Nov. 1994.

Appendix: Proof of Theorem 4

Γq, Γq(u) and B with graded lexicographic order are asdefined in Sect. 4.2.

Lemma A.1 A pair (hi,hj) (hi,hj ∈ B) is well-behaving if and only if fifj = fifj , where f is a reducedpolynomial†of f .(Proof) Assume fifj = fifj . Note that since

hu ∗ hv = ψ(fu)ψ(fv) = ψ(fufv) = ψ(fufv)

and fufv ∈ Γq, we have ρ(hu ∗hv) < ρ(hi ∗hj) for anyhu,hv ∈ B if and only if fufv ≺GL fifj and fufv �=fifj . Thus we show that fufv ≺GL fifj and fufv �=fifj for every u, v such that 1 ≤ u ≤ i, 1 ≤ v ≤ j andu+ v < i+ j.

By the definition of Γq, fu ≺GL fi (resp. fv ≺GL

fj) for 1 ≤ u ≤ i (resp. 1 ≤ v ≤ j) and theequality with respect to ≺GL holds only when u = i(resp. v = j). Since ≺GL is a monomial order, iffu ≺GL fi (resp. fv ≺GL fj) then fufj ≺GL fifj (resp.fufv ≺GL fufj) [5], which implies that

fufv ≺GL fufj ≺GL fifj (A·1)for 1 ≤ u ≤ i and 1 ≤ v ≤ j. In Eq. (A·1), fufv =GL

fifj holds only when u = i and v = j, but which isimpossible for u and v with u+ v < i+ j.

Finally, as f ≺GL f for any monomial in R, wehave fufv ≺GL fifj for u and v which satisfy 1 ≤ u ≤ i,1 ≤ u ≤ j and u + v < i + j. For such u and v,fufv �= fifj is trivial. Therefore it is concluded that(hi,hj) is well-behaving.

Conversely, let fi := X i11 X i2

2 · · ·X imm and fj :=

Xj11 Xj2

2 · · ·Xjmm , and assume that fifj �= fifj . Then

there exists a nonempty set

R := {% : 1 ≤ % ≤ m, i� + j� ≥ q}.For each % ∈ R, let i′� and j′� be integers such that

i′� + j′� = i� + j� − (q − 1),0 ≤ i′� ≤ i�, 0 ≤ j′� ≤ j�.

}

For these i′� and j′�, define

fu :=m∏

�=1

Xu�

� , u� :={

i� for % /∈ R,i′� for % ∈ R,

fv :=m∏

�=1

Xv�

� , v� :={

j� for % /∈ R,j′� for % ∈ R.

Since fu �= fi, fv �= fj and fu ≺GL fi, fv ≺GL fj , wehave 1 ≤ u ≤ i, 1 ≤ v ≤ j and u+ v < i+ j. For thesefu and fv, we also have fufv = fifj , which implies that(hi,hj) is not well-behaving. ✷

Since f ∈ Γq is a reduced monomial, 0 ≤ deg(f) ≤m(q − 1) for all f ∈ Γq.

Definition A.1 For each Λi defined in Eq. (2), wedefine

Λδi := {% ∈ Λi : deg(f�) = δ}, 0 ≤ δ ≤ m(q − 1).

✷

It is obvious that Λi = ∪m(q−1)δ=0 Λδ

i and Λδi (δ =

0, 1, . . . ,m(q − 1)) are mutually disjoint.

Definition A.2 For given fi ∈ Γq, define imax asthe integer such that

fimax = max≺GL{f ∈ Γq : deg(f) = deg(fi)},where maximum is taken with respect to ≺GL. ✷

For given fi ∈ Γq, let Q and R be integers suchthat deg(fi) = Q(q − 1) + R, 0 ≤ Q, 0 ≤ R ≤ q − 2.Then by the definition of graded lexicographic order,fimax is expressed as

fimax = Xq−11 · · ·Xq−1

Q XRQ+1. (A·2)

†A polynomial in R is called reduced if it is a linearcombination of reduced monomials. For every polynomialf ∈ R, there exists a unique reduced polynomial f such thatψ(f) = ψ(f) [2].


By the definition of Λi (Eq. (2)) and Lemma A.1,Λi is rewritten as

Λi = {k : k ∈ {0, 1, 2, . . . , n} such that fk = fifj

where fj ∈ Γq and fifj = fifj}.Moreover, for fi =

∏m�=1 X i�

� , we can write

Λi = {k : k ∈ {0, 1, 2, . . . , n} such that fk = fifj

where fj =∏m

�=1 Xj�

� ,

0 ≤ j� ≤ q − 1− i�, % = 1, 2, . . . ,m}.(A·3)Hereafter, we determine the number i∗ (1 ≤ i∗ ≤

r) for Cr = RMq(u,m) (r = |Γq(u(q − 1) − u − 1)|)which satisfies

|Ai∗ | ≥ |Ai|, for all i, 1 ≤ i ≤ r.

Then gB(Cr) = |Ai∗ | by the definition.

Lemma A.2 For all i ∈ {1, 2, . . . , n},|Λδ

i | ≥ |Λδimax

|, δ = 0, 1, . . . ,m(q − 1).

(Proof) Step 1. At first, we fix i and δ. Let fi =∏m�=1 X i�

� and define the subset Jδi of {0, 1, . . . , q− 1}m

by

Jδi := {(j1, j2, . . . , jm) : 0 ≤ j� ≤ q − 1− i�,

% = 1, 2, . . . ,m,∑m

�=1 j� = δ − deg(fi)}.By Definition A.1 and Eq. (A·3), we see that |Λδ

i | =|Jδ

i |. Thus we show that |Jδi | ≥ |Jδ

imax|.

Fix %1 and %2 (1 ≤ %1 < %2 ≤ m) and define thesubsets Jδ

i (ω) and Jδi (ω, j) of Jδ

i by{Jδ

i (ω) := {(j1, j2, . . . , jm) ∈ Jδi : j�1 + j�2 = ω},

Jδi (ω, j) := {(j1, j2, . . . , jm) ∈ Jδ

i (ω) : j�1 = j}.Since 0 ≤ j�ν ≤ q − 1− i�ν (ν = 1, 2), we have 0 ≤ ω ≤2(q − 1)− (i�1 + i�2) and

Jδi =

2(q−1)−(i�1+i�2 )⋃ω=0

Jδi (ω).

Moreover, Jδi (ω)’s are mutually disjoint for ω =

0, 1, . . . , 2(q − 1)− (i�1 + i�2). Thus

|Jδi | =

2(q−1)−(i�1+i�2)∑ω=0

|Jδi (ω)|. (A·4)

On the other hand, for fixed ω, Jδi (ω, j) is defined

for j such that{0 ≤ j ≤ q − 1− i�1 and0 ≤ ω − j ≤ q − 1− i�2.

(A·5)

Thus

Jδi (ω) =

⋃0 ≤ j ≤ q − 1 − i�1 ,

0 ≤ ω − j ≤ q − 1 − i�2

Jδi (ω, j).

It is obvious that, for fixed ω, Jδi (ω, j)∩Jδ

i (ω, j′) = ∅ forj �= j′, and |Jδ

i (ω, j)|’s are constant for all j satisfyingEq.(A·5). Therefore, |Jδ

i (ω)| can be expressed as

|Jδi (ω)| = N(ω)|Jδ

i (ω, jω)| (A·6)where jω is an arbitrary integer satisfying Eq. (A·5) andN(ω) is the number of j’s satisfying Eq. (A·5) for givenω.

Let σmin := min{q−1−i�1, q−1−i�2} and σmax :=max{q − 1 − i�1 , q − 1 − i�2}. Then it is easily verifiedthat N(ω) is expressed by using i�1 , i�2 and w as

N(ω) =

ω + 1 for 0 ≤ ω ≤ σmin,σmin + 1 for σmin + 1 ≤ ω ≤ σmax − 1,2(q − 1)− (i�1 + i�2)− ω + 1

for σmax ≤ ω ≤2(q − 1)− (i�1 + i�2).

By these expressions and Eqs.(A·4) and (A·6), we have

|Jδi |=

σmin∑ω=0

(ω + 1)|Jδi (ω, jω)|

+σmax−1∑

ω=σmin+1

(σmin + 1)|Jδi (ω, jω)|

+2(q−1)−(i�1+i�2)∑

ω=σmax

ν(ω)|Jδi (ω, jω)| (A·7)

where ν(ω) := 2(q − 1)− (i�1 + i�2)− ω + 1.

Step 2. In this step, we also fix i and δ. Since Eq. (A·7)does not depend on the order of i� in (i1, i2, . . . , im),we assume that i� ≥ i�+1 (% = 1, 2, . . . ,m− 1) when weconsider |Jδ

i |.Let %1 (1 ≤ %1 ≤ m) be the smallest integer such

that i�1 ≤ q − 2 and %2 (1 ≤ %2 ≤ m) be the largestinteger such that i�2 ≥ 1. Since we assume that i� ≥i�+1, we have the following three cases.

(i) The case %1 > %2. This corresponds to(i1, i2, . . . , im) = (q − 1, . . . , q − 1, 0, . . . , 0) wherethe number of q − 1 is Q and Q(q − 1) = deg(fi).

(ii) The case %1 = %2. This corresponds to(i1, i2, . . . , im) = (q−1, . . . , q−1, R, 0, . . . , 0) wherethe number of q − 1 is Q, 1 ≤ R ≤ q − 2 andQ(q − 1) +R = deg(fi).

(iii) The case %1 < %2. All (i1, i2, . . . , im) other thanthe cases (i) and (ii) fall into this case.

We show that if %1 < %2, there exists i′ (�= i) (1 ≤i′ ≤ n) such that deg(fi) = deg(fi′) and |Jδ

i | ≥ |Jδi′ |.

Let

i′� :=

i� for % = 1, 2, . . . ,m, % �= %1, %2,i� + 1 for % = %1,i� − 1 for % = %2,

and fi′ :=∏m

�=1 Xi′�� . Since it holds that 0 ≤ i′� ≤ q−1,


fi′ ∈ Γq. Moreover,

deg(fi′)=m∑

�=1

i′� =m∑

� = 1,

� = �1, �2

i� + (i�1 + 1) + (i�2 − 1)

=m∑

�=1

i� = deg(fi).

For this fi′ , we define Jδi′ , Jδ

i′(ω) and Jδi′ (ω, jω)

similarly to Jδi , J

δi (ω) and Jδ

i (ω, jω) above. It is notedhere that for fixed ω, there exists jω and j′ω for i′�1 andi′�2 which satisfy Eq. (A·5) and |Jδ

i (ω, jω)| = |Jδi′(ω, j′ω)|.

Hence by Eq. (A·7), we have

|Jδi | − |Jδ

i′ |

=q−1−i�1∑

ω=0

(ω + 1)|Jδi (ω, jω)|

+q−2−i�2∑ω=q−i�1

(q − i�1)|Jδi (ω, jω)|

+2(q−1)−(i�1+i�2 )∑

ω=q−1−i�2

ν(ω)|Jδi (ω, jω)|

−q−1−i′�1∑

ω=0

(ω + 1)|Jδi′(ω, j′ω)|

−q−2−i′�2∑ω=q−i′

�1

(q − i′�1)|Jδi′(ω, j′ω)|

−2(q−1)−(i�1+i�2 )∑

ω=q−1−i′�2

ν′(ω)|Jδi′ (ω, j′ω)|

= ((q − 1− i�1) + 1)|Jδi (q − 1− i�1 , jω)|

− (q − (i�1 + 1))|Jδi′ (q − (i�1 + 1), j′ω)|

− (q − (i�1 + 1))|Jδi′ (q − 2− (i�2 − 1), j′ω)|

+ {2(q − 1)− (i�1 + i�2)− (q − 1− i�2) + 1}|Jδ

i (q − 1− i�2 , jω)|

+q−i�2−2∑ω=q−i�1

|Jδi (ω, jω)|

= |Jδi (q − 1− i�1 , jω)|+ |Jδ

i (q − 1− i�2 , jω)|

+q−i�2−2∑ω=q−i�1

|Jδi (ω, jω)|

≥ 0,

where{ν(ω) := 2(q − 1)− (i�1 + i�2)− ω + 1,ν′(ω) := 2(q − 1)− (i′�1 + i′�2)− ω + 1,

Step 3. Let %1 and %2 be integers defined in Step 2.

For given fi =∏m

�=1 X i�

� , if there exists no i′ (�= i)such that deg(fi) = deg(fi′) and |Jδ

i | ≥ |Jδi′ |, then by

Step 2, we can conclude that %1 ≥ %2. This means(i1, i2, . . . , im) = (q−1, . . . , q−1, R, 0, . . . , 0) where thenumber of q − 1 is Q, 0 ≤ R ≤ q − 2 and deg(fi) =Q(q − 1) +R. Therefore by Eq. (A·2), fi = fimax .

Finally, by noting that the above discussion holdsfor any i and δ, we have the lemma. ✷

Lemma A.3 For i, j ∈ {1, 2, . . . , n} such thatdeg(fi) < deg(fj), we have Λimax ⊃ Λjmax .(Proof) It is sufficient to prove it for the case whendeg(fi) + 1 = deg(fj). By Eqs. (A·2) and (A·3), Λimax

is rewritten as

Λimax = {k : k ∈ {0, 1, . . . , n} s.t fk = fimaxfu

where fu = XuQ+1Q+1 X

uQ+2Q+2 · · ·Xum

m ,

0 ≤ uQ+1 ≤ q − 1−R, 0 ≤ u� ≤ q − 1for % = Q+ 2, Q+ 3, . . . ,m}.

Note that by changing the condition on uQ+1 as 0 ≤uQ+1 ≤ q − 2 − R, we obtain the exactly similar ex-pression for Λjmax and these expressions for Λimax andΛjmax immediately imply Λimax ⊃ Λjmax . ✷

(Proof of Theorem 4) By noting that deg(fr+1) =deg(fr) + 1, we can write

Λi =

deg(fr)⋃

δ=0

Λδi

⋃

m(q−1)⋃

δ=deg(fr+1)

Λδi

.

Since deg(f�) < deg(fr+1) for % = 1, 2, . . . , r, we havedeg(fr)⋃

δ=0

Λδi

⋂{r + 1, r + 2, . . . , n} = ∅.

On the other hand,

Λi :=

m(q−1)⋃

�=deg(fr+1)

Λδi

⊂ {r + 1, r + 2, . . . , n}.

Thus

Ai = {r + 1, r + 2, . . . , n} \ Λi

= {r + 1, r + 2, . . . , n} \ Λi.

and |Ai| = n− r − |Λi|.For i ≤ r, we have from Lemma A.2 that

|Λi|=m(q−1)∑

δ=deg(fr+1)

|Λδi |

≥m(q−1)∑

δ=deg(fr+1)

|Λδimax

| = |Λimax |.

Moreover, for i ≤ r,


|Λimax | ≥ |Λrmax | = |Λr|by Lemma A.3. Thus |Λi| ≥ |Λr| for all 1 ≤ i ≤ r.

Finally, |Λr| is equal to the number of (j1, j2,. . . , jm) �= (0, 0, . . . , 0) satisfying

j� = 0 for % = 1, 2, . . . , Q,0 ≤ jQ+1 ≤ q − 1−R,0 ≤ j� ≤ q − 1 for % = Q+ 2, Q+ 3, . . . ,m.

Thus |Λr| = (q −R)qm−(Q+1) − 1 and we have

gB(Cr)=n− r − |Λr|=n− r − (q −R)qm−(Q+1) + 1.

✷

Tomoharu Shibuya received B.E.,M.E., and Ph.D. degrees in electrical andelectronic engineering from Tokyo Insti-tute of Technology, Tokyo, Japan, in1992, 1994 and 1999, respectively. Since1994, he has been a research associate inthe Department of Electrical and Elec-tronic Engineering, Tokyo Institute ofTechnology. His research interests are er-ror control coding and information theory.

Ryo Hasegawa received the B.E. de-gree in Computer Science from Tokyo In-stitute of Technology in 1998. Since 1998,he is a researcher at Tokyo Institute ofTechnology. His research interest is cod-ing theory.

Kohichi Sakaniwa received B.E.,M.E., and Ph.D. degrees all in electronicengineering from the Tokyo Institute ofTechnology, Tokyo Japan, in 1972, 1974and 1977, respectively. He joined the To-kyo Institute of Technology in 1977 as aresearch associate and served as an asso-ciate professor from 1983 to 1991. Since1991 he has been a professor in the De-partment of Electrical and Electronic En-gineering in the Tokyo Inst. of Tech. He

received the Excellent Paper Award from the IEICE (the Insti-tute of Electronics, Information and Communication Engineers)of Japan in 1982, 1990, 1992 and 1994. His research area in-cludes Communication Theory, Error Correcting Coding, (Adap-tive) Digital Signal Processing and so on. Dr. Sakaniwa is amember of IEEE, Information Processing Society of Japan, In-stitute of Television Engineers of Japan and Institute of ElectricalEngineers of Japan.

paper a lower bound for generalized hamming weights ...the generalized hamming weights, which is...

Documents