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Metrics that respect support
Roberto Assis Machado
University of CampinasUniversity of Illinois at Urbana-Champaign
July 26th 2018
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Summary
1 Introduction
2 Conditional Sums of known weights
3 Representation of weights by directed graphs
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Distance & Coding
Distances are in the realm of coding: 1
Figura: Huffman & Pless Fundamentals of Error-Correcting Codes
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Distance & Coding
Distances are in the realm of coding: 2
Figura: MacWilliams & Sloane Theory of Error-Correcting Codes , page 6.
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Distance & Coding
Distances are in the realm of coding: 3
Figura: Ling & Xing Coding Theory, a First Course .
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The Hamming Metric
The big star
Given u, v ∈ Fnq, with u = (u1, u2, . . . , un), v = (v1, v2, . . . , vn)
dH(u, v) = |{i : ui 6= vi}|.
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What makes the Hamming distance so special
It fits a Symmetric Channel
It matches the Binary Symmetric Channel:
dH(u, v) ≤ dH(u,w)
if and only if
P(v = received|u = sent) ≥ P(w = received|u = sent).
This implies that, for every given code C ⊆ Fnq, decoding using maximal
likelihood or minimal distance leads to the same result:
arg maxv∈C
Pr(u = received | v = sent) = arg minv∈C
d(u, v). (1)
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Invariant metrics
Such matching allowed to explore the code properties as invariant metrics:
packing radius;
covering radius;
MacWilliams’ Identity;
equivalence of codes;
and so forth.
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Metrics introduced in Coding Theory
Through the years, many metrics were introduced in the context of codingtheory.
Poset metrics of Brualdi [2];
Gabidulin’s combinatorial metrics [6];
Poset-block metrics [1];
Digraph metrics [5].
Shared properties
P1: Weight condition, i.e., d(u, v) = wt(u − v).
P2: Support condition, i.e., wtH(u) ≤ wtH(v), if supp(u) ⊂ supp(v). Inthis case, we say that the metric respects support.
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Metrics introduced in Coding Theory
Through the years, many metrics were introduced in the context of codingtheory.
Poset metrics of Brualdi [2];
Gabidulin’s combinatorial metrics [6];
Poset-block metrics [1];
Digraph metrics [5].
Shared properties
P1: Weight condition, i.e., d(u, v) = wt(u − v).
P2: Support condition, i.e., wtH(u) ≤ wtH(v), if supp(u) ⊂ supp(v). Inthis case, we say that the metric respects support.
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Weight function
A map wt : Fnq → Z is a weight function if given u, v ∈ Fn
q:
wt(u) ≥ 0, and wt(u) = 0 ⇐⇒ u = 0.
supp(u) ⊂ supp(v)⇒ wt(u) ≤ wt(v), where
supp(u) = {i ∈ {1, 2, . . . , n} : ui 6= 0},
in this case we say that the weight function respects support.
A weight function wt determines a semi-metric by definingd(u, v) = wt(u − v).
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Weight function
A map wt : Fnq → Z is a weight function if given u, v ∈ Fn
q:
wt(u) ≥ 0, and wt(u) = 0 ⇐⇒ u = 0.
supp(u) ⊂ supp(v)⇒ wt(u) ≤ wt(v), where
supp(u) = {i ∈ {1, 2, . . . , n} : ui 6= 0},
in this case we say that the weight function respects support.
A weight function wt determines a semi-metric by definingd(u, v) = wt(u − v).
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Weight function
A map wt : Fnq → Z is a weight function if given u, v ∈ Fn
q:
wt(u) ≥ 0, and wt(u) = 0 ⇐⇒ u = 0.
supp(u) ⊂ supp(v)⇒ wt(u) ≤ wt(v), where
supp(u) = {i ∈ {1, 2, . . . , n} : ui 6= 0},
in this case we say that the weight function respects support.
A weight function wt determines a semi-metric by definingd(u, v) = wt(u − v).
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Example
00 10
01 11
00 10
01 11
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Possible decoding criteria in F22
Criterion Hamming Poset Poset-block Combinatorial
wt(10) = wt(01) < wt(11) X X X Xwt(10) = wt(01) = wt(11) X Xwt(10) < wt(01) = wt(11) X Xwt(10) < wt(01) < wt(11)
The decoding criteria 0 = wt(00) < wt(10) < wt(01) < wt(11) is notcovered by any of such known metrics.
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Possible decoding criteria in F22
Criterion Hamming Poset Poset-block Combinatorial
wt(10) = wt(01) < wt(11) X X X Xwt(10) = wt(01) = wt(11) X Xwt(10) < wt(01) = wt(11) X Xwt(10) < wt(01) < wt(11)
The decoding criteria 0 = wt(00) < wt(10) < wt(01) < wt(11) is notcovered by any of such known metrics.
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How to explore weight functions
In order to understand the space of all metrics satisfying conditions P1 andP2, it is enough to study the space of all weight functions that respectsupport.There are two approaches to explore such a family:
By conditional sums of known metrics;
By representing each weight function as an edge-weighted directedgraph.
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Conditional Sums of known weights
Linear isometries
Let wt1 and wt2 be weights respecting support. Then,
1 (Direct sum:) wt1 + wt2 respects support;
2 (k-sum:) wt1 ⊕k wt2 respects support, where
(wt1 ⊕k wt2)(u) =
{wt1(u), if wt1(u) < k ,
wt1(u) + wt2(u), if wt1(u) ≥ k .
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Conditional sum reaches new decoding criteria
Criterion Hamming Poset Poset-block Combinatorial
wt(10) = wt(01) < wt(11) X X X Xwt(10) = wt(01) = wt(11) X Xwt(10) < wt(01) = wt(11) X Xwt(10) < wt(01) < wt(11)
The decoding criteria 0 = wt(00) < wt(10) < wt(01) < wt(11) can bereached by
wt(x) = wt2(x)⊕2 wtH(x) =
{wt2(x), if |supp(x)| < 2,
wtH(x) + wt2(x), if |supp(x)| = 2.,
where wt2(00) = 0, wt2(10) = 1 and wt2(01) = wt2(11) = 2 (a posetmetric).
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Conditional sums of known weights
In a more general setting, the k condition can be replaced by anycondition C respecting support.
wt(x) = wt1(x)⊕C wt2(x) =
{wt1(x) + wt2(x), if C ,
wt1(x), otherwise.
C respecting support
A condition C respects support if supp(u) ⊂ supp(v) and u satisfies acondition C , then v also does.
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Conditional sums of known weights
Classes of equivalences for conditional sums
Let wt be a weight function. Then, wt⊕C wt and wt are equivalent if,and only if, ⊕C
∼= ⊕k , for some k ∈ N.
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Representation of weights by directed graphs
Consider an edge-weighted directed graph G (V ,E , δ), where
V = Fn2;
E = {(u, v) : supp(u) ⊂ supp(v) and dH(u, v) = 1};δ : E → Z.
So that, wt(w) =∑
(u,v)∈T δ((u, v)), where T is a trail from the nullvector to w .
Hamming weight
10 01
00
11
1 1
11
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Every weight function admits a representation
Given a weight wt : Fnq → N, δ is defined as by δ((u, v)) = wt(v)− wt(u)
for every v = u + ei ∈ Fnq.
10 01
00
11
wt(10)
wt(11) − wt(01)
wt(01)
wt(11) − wt(10)
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Constraints on δ to construct a weight function
To ensure that a δ-digraph induces a weight it is required some extraconstraints on δ.
10 01
00
11
1 0
11
δ((0, ei ) > 0;
The summation wt(w) =∑
(u,v)∈T δ((u, v)) can not depend on thetrail.
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Classification of Gabidulin’s combinatorial metrics
Proposition
Let G (Fn2,E , δ) be a δ-digraph determined by a weight function wt. Then,
wt is combinatorial if, and only if, δ((u, v)) ∈ {0, 1}.
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Linear Isometries
Induced by permutations
Let φ ∈ Sn be a permutation.fφ(x1, . . . , xn) = (xφ(1), . . . , xφ(n)) is a linear isometry if, and only if, fφ is aδ-digraph automorphism.
10 01
00
11
1 1
11
10 01
00
11
1 2
12
Permutation is linear isometry Permutation is not linear isometry
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Linear Isometries
Induced by permutations
Let φ ∈ Sn be a permutation.fφ(x1, . . . , xn) = (xφ(1), . . . , xφ(n)) is a linear isometry if, and only if, fφ is aδ-digraph automorphism.
10 01
00
11
1 1
11
10 01
00
11
1 2
12
Permutation is linear isometry Permutation is not linear isometry
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Linear Isometries
Increasing the support
Consider a map f :Fnq→Fn
q defined as f (ei ) =∑n
j=1 λjej , with λi 6= 0. If∑(x ,y)∈T δ((x , y)) = 0 for every trail from u ∈ Fn
2 with ui 6= 0 tou + f (ei )− ei , then f is a linear isometry.
0
e1 e2 e3
e1 + e2e1 + e3
e2 + e3
e1 + e2 + e3
1 1 1
0 0 0101
10
1
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Linear Isometries
Increasing the support
Consider a map f :Fnq→Fn
q defined as f (ei ) =∑n
j=1 λjej , with λi 6= 0. If∑(x ,y)∈T δ((x , y)) = 0 for every trail from u ∈ Fn
2 with ui 6= 0 tou + f (ei )− ei , then f is a linear isometry.
0
e1 e2 e3
e1 + e2e1 + e3
e2 + e3
e1 + e2 + e3
1 1 1
0 0 0101
10
1
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References
Marcelo Muniz Silva Alves, Luciano Panek, Marcelo Firer, et al.
Error-block codes and poset metrics.Advances in Mathematics of Communications, 2008.
Richard A. Brualdi, Janine Smolin Graves, and K.Mark Lawrence.
Codes with a poset metric.Discrete Mathematics, 147(1 - 3):57 – 72, 1995.
Michel Marie Deza and Elena Deza.
Encyclopedia of distances.In Encyclopedia of Distances, pages 1–583. Springer, 2009.
Rafael Gregorio Lucas D’Oliveira and Marcelo Firer.
Channel metrization.CoRR, abs/1510.03104, 2015.
T. Etzion, M. Firer, and R. A. Machado.
Metrics based on finite directed graphs and coding invariants.IEEE Transactions on Information Theory, PP(99):1–1, 2017.
EM Gabidulin.
Combinatorial metrics in coding theory.In 2nd International Symposium on Information Theory. Akademiai Kiado, 1973.
EM Gabidulin.
A brief survey of metrics in coding theory.Mathematics of Distances and Applications, 66, 2012.
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Gabidulin’s Combinatorial metric
Basic settings, definitions and notation
Pn := {A : A ⊂ {1, 2, . . . , n}};Given X ⊂ {1, 2, . . . , n}, a family F ⊂ Pn is a covering of X ifX ⊂ ∪A∈FA;
We assume F is a covering of {1, 2, . . . , n}.
Definition: Combinatorial weight
Given u = (u1, . . . , un) ∈ Fnq, the F-combinatorial weight is defined by
wtF (x) = min{|A| : A ⊂ F and A is a covering of supp(x)}.
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