IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 6, SEPTEMBER 1999 2207

Since�K(g) and�L(g) equal the number of elements�K(g) and�L(g) fix the following character inner products are easily computed

1 =1

jGjg2G

�K(g)�3(g)

=1

jGjg2G

�K(g)�4(g)

1 =1

jGjg2G

�L(g)�1(g)

=1

jGjg2G

�L(g)�2(g)

=1

jGjg2G

�L(g)�4(g):

Now, since every permutation representation contains the identityrepresentation we have,�K = 1��3��4 and�L = 1��1��2��4.So we can represent group codes generated by�1 and �2 byequivalent permutation codes but the degree would be12 and thematrix V would have to be found.

We conclude with an example which summarizes the main resultof the correspondence.

Example 2.4: Let G be the icosahedral group. Consider the four-dimensional group codeX = f�3(g)xxx: g 2 Gg. The image of�3can be found in [7, p. 313]. We use a modification of the algorithmin (5) to find the optimal initial vector

xxx = [�0:68222; �0:49471; �0:44657; �0:30070]

for this representation. The minimum squared Euclidean distance isd2min = 0:447056. We are then under the hypothesis of Corollary2.1. We use the degree5 permutation representation�H = 1 � �3and transform the zero-padded vector

xxx0 = [0; �0:68222; �0:49471; �0:44657; �0:30070]

to the initial vector

VTxxx0 = [�0:61010; �0:27588; �0:06926; 0:26504; 0:69020]

for the equivalent permutation code generated by�H.We finally note that in practice the code is transmitted over the

additive white Gaussian noise (AWGN) channel using the low-dimensional constellation in order to save on the spectral efficiency.The received vectorrrr is first zero-padded as for the initial vectorand then transformed intoyyy = V Trrr0. Now, yyy can be maximum-likelihood (ML) decoded with the permutation code decoder. We notethat this is an orthogonal transformation on the received vector whichdoes not modify the additive noise statistics. In the above examplethe operation is particularly convenient since the code dimension isonly increased by one. On the other hand, if we wanted to use thethree-dimensioanl codes generated by the representations�1 or �2we would need to use a degree12 permutation representation.

REFERENCES

[1] E. Biglieri and M. Elia, “Cyclic-group codes for the Gaussian channel,”IEEE Trans. Inform. Theory, vol. IT-22, pp. 624–629, Sept. 1976.

[2] I. F. Blake, “Distance properties of group codes for the Gaussianchannel,”SIAM J. Appl. Math., vol. 23, no. 3, pp. 312–324, Nov. 1972.

[3] C. W. Curtis and I. Reiner, “Representation theory of finite groups andassociative algebras,”Interscience, New York, 1962.

[4] J. K. Karlof, “Decoding spherical codes for the Gaussian channel,”IEEETrans. Inform. Theory, vol. 39, pp. 60–65, Jan. 1993.

[5] , “Permutation codes for the Gaussian channel,”IEEE Trans.Inform. Theory, vol. 35, pp. 726–732, July 1989.

[6] R. Kochendorffer, Group Theory. London, U.K.: McGraw-Hill, 1965.[7] J. S. Lomont,Applications of Finite Groups. New York: Academic,

1959.[8] V. M. Sidelnikov, “On a finite group of matrices generating orbit codes

on Euclidean sphere,” inProc. 1997 IEEE Int. Symp. Information Theory(Ulm, Germany, June 29–July 4, 1997), p. 436.

[9] D. Slepian, “Permutation modulation,”Proc. IEEE, vol. 53, pp.228–236, Mar. 1965.

[10] , “Group codes for the Gaussian channel,”Bell Syst. Tech. J., vol.47, pp. 575–602, Apr. 1968.

On Double-Byte Error-Correcting Codes

C.-L. Chen

Abstract—This correspondence shows that there is a flaw in the resultspresented in [1]. A large family of the codes constructed in [1] are notdouble-byte error-correcting codes as originally claimed.

Index Terms—Double-byte error correction, error-correcting code.

I. INTRODUCTION

The authors in [1] claim that a class of double-byte error-correctingcodes over GF(q), q a power of2, has been constructed. These codeshave the parameters of code lengthn = qm and code redundancyr � 2m + m

3+ 1, for any integerm equal to or greater than3.

However, a close examination of the paper reveals a flaw in the proofsof theorems that renders the claim invalid. This correspondence is topoint out the flaw in [1].

Five constructions of linear codes have been presented in [1]. Eachconstruction has been claimed to have a minimum distance of five andthus have the ability to correct all double-byte errors. Constructions3.2 and 5.1 are equivalent to cyclic codes extended by one byte.The minimum distance of these codes can be shown to be five ormore by counting multiple sets of consecutive roots in their generatorpolynomials [2]. The codes constructed from Construction 3.1 areshortened codes of those constructed from Construction 3.2. Theminimum distance of these codes is at least five. The large family ofcodes constructed from Constructions 3.3 and 3.4 are not double-byteerror-correcting codes as claimed by the authors. Counter exampleswill be presented in the next section to show that codes constructedaccording to Constructions 3.3 and 3.4 contain codewords of weightfour and thus do not have a double-byte error-correcting ability.

II. COUNTER EXAMPLES

The propositions of Theorems 3.3 and 3.4 in [1] state that thecodes obtained from Constructions 3.3 and 3.4 in [1] have a min-imum distance of five or more. In this section, two examples ofConstructions 3.3 and 3.4 are presented. It will be shown that the

Manuscript received August 25, 1998.The author is with IBM Corporation, MS P361, Poughkeepsie, NY 12601

USA.Communicated by A. M. Barg, Associate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(99)05861-7.

0018–9448/99$10.00 1999 IEEE

2208 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 6, SEPTEMBER 1999

codes constructed have codewords of weight four. Therefore, thepropositions of Theorems 3.3 and 3.4 of [1] are invalid. In bothexamples, the setf0; 1; �; �2g represents the elements of GF(q),q = 22, and� represents a root of the polynomial� + x+ x2 + x3.Note that� 2 GF(q3), andf1; �; �2g is a basis of the vector spaceGF(q3) over GF(q).

Example 1: Let be a root of the polynomial�+�x+x2+x3+x4.Then, 2 GF(q4), andf1; ; 2; 3g is a basis of the vector spaceGF(q4) over GF(q). An element i of GF(q4) can be expressed asa polynomial of over GF(q), i.e., i = xi + yi + zi

2 + !i 3,

where the coefficients of the powers of are elements of GF(q).Let H be a matrix ofn = q4 distinct columns, of which a columnhi is defined as

hi = [1; i;

i(q+1); (xi + yi� + zi�

2)q +q+1; !

q +q+1i ]T :

Then the linear code of lengthn over GF(q) havingH as a parity-check matrix is constructed according to Construction 3.3 of [1].

Now consider columns ofH for i = 79; 130; 164; and215. Then

h79 = [1; �2 + + �2 2; 1 + �

2 + �

2 2 +

3; �; 0]T

h130 = [1; �2 + � ; 1 + �2 + �

2 2 +

3; �; 0]T

h164 = [1; 1 + � + 2; �

2 + � + � 2 + �

2 3; �; 0]T

h215 = [1; 1 + �2 ; �

2 + � + � 2 + �

2 3; �; 0]T

and �h79 + �h130 + h164 + h215 = 0. Thush79; h130; h164; andh215 are dependent over GF(q) and there is a codeword of weightfour in the code [3].

Example 2: Let be a root of the polynomial�+x+x2+x6. Then, 2 GF(q6) andf1; ; 2; 3; 4; 5g is a basis of the vector spaceGF(q6) over GF(q). An element i of GF(q6) can be expressed asa polynomial of , i.e.,

i = xi0 + xi1 + xi2

2 + xi3 3 + xi4

4 + xi5 5

with the coefficients in GF(q). Let H be a matrix ofn = q6 distinctcolumns, of which a columnhi is of the form

hi = [1; ; q+1; (xi0 + xi1� + xi2�2)q +q+1

;

(xi3 + xi4� + xi5�2)q +q+1]T :

Then the linear code of lengthn over GF(q) havingH as a parity-check matrix is constructed according to Construction 3.4 of [1].

Now, consider the following four columns ofH:

h359 = [1; �2 + + � 3 + �

2 4; � +

2 + �2 3; �; �]T

h907 = [1; 1 + + � 3; � + �

2 2 + �

2 3 + �

2 4; �

2; 1]T

h1288 = [1; �2 + + 3 +

4;

�2 + �

2 2 + �

3 + 4 +

5; �; �

2]T

h1518 = [1; 1 + + �2 3 + �

2 4;

3 + 4 + �

5; �

2; �

2]T :

Since�h359 + h907 + �h1288 + h1518 = 0, there is a codeword ofweight four in the code.

III. D ISCUSSION

The examples in Section II show that Theorems 3.3 and 3.4 of [1]are false. The flaw in the proofs of these two theorems in [1] is inthe degenerate cases of the parameterc and the function(x+ cy)q+1

that the authors failed to consider.In the proof of Theorem 3.3, the authors derive equation (3.6) from

equation (3.5) and from the expressions

cq + c = m0 +m1 +m2

2 +m3 3

cq+1 = n0 + n1 + n2

2 + n3 3

(x+ cy)q+1 = x2 +m0xy + y

2 + (m1xy + y2)

+ (m2xy + y2) 2 + (m3xy + y

2) 3:

However, the authors failed to recognize that some of the parametersmay be null. The mistake is similar to the one spotted in [5] bythe authors in [4]. In particular, if(x + cy)q+1 has a constantterm only, then three of the four equations in (3.6) that involvethe xy term would disappear and Lemma 3.1 cannot be appliedto prove Theorem 3.3. This is the case in Example 1, which haszi = �2xi + yi, for i = 79; 130; 164;215; andcq + c = cq+1 = �2.Thus (x + cy)q+1 = x2 + �2xy + y2, which is a constant withrespect to .

Similarly, in the proof of Theorem 3.4, the authors failed toconsider the degenerate cases of(x + cy)q+1 as a function of .If the function is a constant, then some of the equations in (3.11) in[1] vanish. Again, Lemma 3.1 becomes inappropriate to prove that(3.11) cannot have four roots, and the minimum distance propositionof Theorem 3.4 cannot be verified.

REFERENCES

[1] G. L. Feng, X. Wu, and T. R. N. Rao, “New double-byte error-correctingcodes for memory systems,”IEEE Trans. Inform. Theory, vol. 44, pp.1152–1163, May 1998.

[2] C. R. P. Hartmann and K. K. Tzeng, “Generalizations of the BCHbound,” Inform. Contr., vol. 20, pp. 489–498, June 1972.

[3] E. J. Weldon, Jr and W. W. Peterson,Error-Correcting Codes, 2nd ed.Cambridge, MA: MIT Press, 1972.

[4] I. M. Duursma and R. K¨otter, “Error-locating pairs for cyclic codes,”IEEE Trans. Inform. Theory, vol. 40, 1108–1121, July 1994.

[5] G. L. Feng and K. K. Tzeng, “Decoding cyclic and BCH codes upto actual minimum distance using nonrecurrent syndrome dependencerelations,” IEEE Trans. Inform. Theory, vol. 37, pp. 1716–1723, Nov.1991.