IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
which implies from (A.2) and (A.3)
A
k ( x; y; a ) =
0kA (x; y; b):
(A.6)
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Some Bounds for the Minimum Length of Binary Linear Codes of Dimension Nine Iliya Bouyukliev, Sugi Guritman, and Vesselin Vavrek
REFERENCES [1] P. Elias, “Coding for noisy channels,” in IRE Conv. Rec., vol. IRE-2, 1955, pp. 37–47. [2] A. J. Viterbi, “Error bounds for convolutional codes and asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 260–269, Apr. 1967. [3] M. P. C. Fossorier, S. Lin, and D. Rhee, “Differential Trellis Decoding for Convolutional Codes,” LSI Logic Corp., US Patent 5 781 569, July 1998. [4] T. Hardin and S. Gardner, “Accelerating Viterbi decoder simulations,” Commun. Syst. Des., vol. 5, pp. 52–58, Jan. 1999. [5] G. D. Forney Jr., “Coset codes II: Binary lattices and related codes,” IEEE Trans. Inform. Theory, vol. 34, pp. 1152–1187, Sept. 1988. [6] S. Lin and D. J. Costello Jr., Error Control Coding Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1983. [7] Y. Berger and Y. Be'ery, “Soft trellis-based decoder for linear block codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 764–773, May 1994. [8] M. P. C. Fossorier, “Dynamic threshold detection for maximum likelihood sequence estimation,” IEEE Trans. Commun., vol. 44, pp. 1444–1454, Nov. 1996. [9] J. B. Cain, G. C. Clark Jr., and J. M. Geist, “Punctured convolutional codes of rate (n 1)=n and simplified maximum likelihood decoding,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 97–100, Jan. 1979. [10] D. Haccoun and G. Beguin, “High-rate punctured convolutional codes for Viterbi and sequential decoding,” IEEE Trans. Commun., vol. 37, pp. 1113–1125, Nov. 1989. [11] G. Beguin, D. Haccoun, and C. Paquin, “Further results on high-rate punctured convolutional codes for Viterbi and sequential decoding,” IEEE Trans. Commun., vol. 38, pp. 1922–1928, Nov. 1990. [12] J. K. Omura, “On the Viterbi decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 177–179, Jan. 1969. [13] J. Snyders and Y. Be'ery, “Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes,” IEEE Trans. Inform. Theory, vol. 35, pp. 963–975, Sept. 1989. [14] J. K. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 76–80, Jan. 1978. [15] J. H. Conway and N. J. A. Sloane, “Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 41–50, Jan. 1986. [16] M. Ran and J. Snyders, “On maximum likelihood decoding of some binary self-dual codes,” IEEE Trans. Commun., vol. 41, pp. 439–446, Mar. 1993. [17] M. P. C. Fossorier and S. Lin, “Coset codes viewed as terminated convolutional codes,” IEEE Trans. Commun., vol. 44, pp. 1096–1106, Sept. 1996. [18] O. Amrani, Y. Be'ery, A. Vardy, F. W. Sun, and H. C. A. van Tilborg, “The Leech lattice and the Golay code: Bounded-distance decoding and multilevel constructions,” IEEE Trans. Inform. Theory, vol. 40, pp. 1030–1043, July 1994. [19] A. Vardy and Y. Be'ery, “More efficient soft decoding of the Golay codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 667–672, May 1991.
0
Abstract—We prove the nonexistence of binary [69 9 32] codes and construct codes with parameters [76 9 34] [297 9 146] and [300 9 148]. These results show that (9 32) = 70, (9 34) 76, (9 146) = 297, and (9 148) = 300, where ( ) denotes the smallest value of for which there exists an [ ] binary code. We also present some codes of minimum distance 32 and some related codes. Index Terms—Minimum-length bounds, optimal binary linear codes.
I. INTRODUCTION
Let F2n be the n-dimensional vector space over the Galois field
. The Hamming distance between two vectors of F2n is defined to be the number of coordinates in which they differ. A binary linear n [n; k; d]-code is a k -dimensional linear subspace of F2 with a minimum distance d. Let n(k; d) denote the smallest value of n for which an [n; k; d]-code exists. An [n(k; d); k; d] code is called optimal. One of the central problems in algebraic coding theory is to determine n(k; d) for all values of k and d. A general lower bound on n(k; d) has been proved by Griesmer [9]. He showed that F2
n(k; d)
k g (k; d)
01
= i=0
d
2i
where dxe denotes the smallest integer x. Baumert and McEliece [1] have shown that for fixed k there exists a number D(k) such that n(k; d) = g (k; d) if d D(k). So for fixed k all but a finite number of values of n(k; d) are known. Let us discuss the situation for small values of k . Baumert and McEliece [1] determined n(k; d) for k 6, van Tilborg [13] completed the case of k = 7 and Bouyukliev, Jaffe, and Vavrek [2] completed the case of k = 8. Although in recent years much effort has been spent on n(9; d), still many values remain unknown. Considerable progress has been made after the works of Dodunekov, Encheva, and Ivanov [7], Bhandari and Garg [6], and many others (refer to Brouwer [4]). Recently [8] Dodunekov, Guritman, and Simonis proved the nonexistence of codes of parameters [132; 9; 64]; [243; 9; 120]; [251; 9; 124]; [370; 9; 184]; and constructed codes of parameters [319; 9; 158]; [315; 9; 156]; and [312; 9; 154]. This correspondence contains four improvements of n(9; d). In Section II we prove the nonexistence of a code with parameters [69; 9; 32], i.e., n(9; 32) = 70. In Section III we present some new codes and thus prove that n(9; 34) 76, n(9; 146) = 297, and n(9; 148) = 300. Section IV presents some optimal codes of minimum distance 32 and their related codes.
Manuscript received March 31, 1999; revised October 8, 1999. The work of I. Bouyukliev was supported in part by the UNESCO UVO-ROSTE under Contract 875.630.9. I. Bouyukliev and V. Vavrek are with the Institute of Mathematics, Bulgarian Academy of Sciences, 5000 V. Tarnovo, Bulgaria. S. Guritman is with the Delft University of Technology, Faculty of Information Technology and Systems, Department of Pure Mathematics, 2600 GA Delft, The Netherlands. Communicated by A. M. Barg, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(00)02895-9. 0018–9448/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
II. NONEXISTENCE OF [69; 9; 32] CODES It is known that Theorem 1 [10], [11]: A putative [69; 9; 32]-code has the unique weight distribution
A0 = 1; A32 = 351; A40 = 156; A48 = 4; B2 = 0:
(1)
Definition: Let C be an [n; k; d]-code over F2 , G be a generator matrix of C , and let c 2 C be a codeword of weight w . Then the residual code of C with respect to c, denoted by Res (C ; c), is the code generated by the restriction of G to the columns where c has a zero entry. If only the weight w of c is of importance we will denote it by Res (C ; w). Lemma 1 [7]: Let C be an [n; k; d]-code and let d > (w=2). Then w) has the parameters [n 0 w; k 0 1; d 0 b(w=2)c].
Res (C ;
A linear code is called projective if no two columns of a generator matrix are linearly dependent. Lemma 2: Let C69 be a [69; 9; 32]-code. Then Res (C69 ; 48) is a projective doubly even code with parameters [21; 8; 8]. There exists a unique such code C21 . Proof: It follows from Theorem 1 and Lemma 1 that Res (C69 ; 48) is doubly even, projective with minimum distance 8. Jaffe [11] classified the even codes of parameters [21; 8; 8]. He proved that there are eight such codes with different weight distributions. Only three of them are doubly even and only one of the three is projective with weight distribution A0 = 1; A8 = 102; A12 = 144; A16 = 9: Let us try to construct a generator matrix G69 of a putative [69; 9; 32]-code C69 with a weight distribution as in Theorem 1. So there are four codewords of weight 48 in C69 . Let ui 2 C; wt (ui ) = 48; i = 1; 2; 3; 4. Then by Lemma 2 for i; j 2 f1; 2; 3; 4g; i 6= j we have wt (uj )
0 wt (u 3 u ) = 48 0 (48 0 (1=2) wt (u + u )) j
i
i
j
= (1=2) wt (ui + uj ) = 16
where u 3 v = ( 1 1 ;
111;
u = ( 1 ; 1 1 1 ; n );
n
n)
for
v = ( 1 ; 1 1 1 ; n ) 2 F2n :
This implies wt (ui + uj ) = 32 and wt (ui 3 uj ) = 32. Similarly, we prove that wt (u1 + u2 + u3 ) = 32. Hence, the vectors u1 ; u2 ; u3 ; u4 are linearly independent. We can fix the first three vectors u1 ; u2 ; u3 without loss of generality, but for the fourth vector u4 we obtain two inequivalent possibilities; u04 and u400 . Let u1 ; u2 ; u3 ; u4 be the first four rows of G69 , where u4 = u40 and u4 = u400 . We will try to fix a generator matrix of the code Res (C69 ; u1 ) depending on the vectors u1 ; u2 ; u3 ; u4 . The vectors v1 ; v2 ; 1 1 1 ; v9 are the weight 16 vectors of C21 111111001111111101100v1 111010111111110110110v2 110101111111111011010v3 111111110110110001111v4 110011111001111101111v5
111101011101100111111v6 100110011110111111111v7
111110101101011011111v8
011001101110111111111v9:
Let Aut (C21 ) be the automorphism group of the code C21 . Using a computer we obtain that the order of Aut (C21 ) is 432. To prove the next lemmas, we use the following permutations from Aut (C21 ):
1 = (1; 2; 9) (3; 16; 8; 7; 15; 4) (5; 6; 17) (10; 19; 13; 20; 14; 18) (11; 21) 2 = (1; 2; 18; 9; 13; 14; 19; 10) (3; 4; 12; 6; 15; 16; 11; 5) (7; 21; 17; 8) 3 = (2; 19; 20; 9; 14; 13) (3; 21; 17; 15; 11; 6) (5; 8; 12) (7; 16) (10; 18) 4 = (2; 19; 10) (4; 12; 5) (6; 16; 11) (7; 21; 17) (9; 14; 18) 5 = (3; 6; 7) (4; 8; 5) (10; 13; 14) (15; 17; 16) (18; 20; 19) 6 = (3; 6) (5; 8) (10; 18) (11; 21) (13; 19) (14; 20) (15; 17) Lemma 3: All vectors of weight 16 in C21 are in the same orbit with respect to Aut (C21 ). Proof: 1 (v1 ) = v6 , 1 (v6 ) = v2 , 1 (v2 ) = v5 , 1 (v5 ) = v3 , 1 (v3 ) = v8 , 1 (v4 ) = v9 , 1 (v9 ) = v7 , 2 (v4 ) = v8 . Lemma 4: Let v be a vector of weight 16 in C21 . All the other vectors of weight 16 in C21 form a unique orbit with respect to St (v ) Aut (C21 ). Proof: Using Lemma 3 we can take v = v9 without loss of generality. The order of St (v ) is 432=9 = 48. The permutations 3 and 4 fix the vector v and 3 (v1 ) = v4 , 3 (v4 ) = v6 , 3 (v6 ) = v8 , 3 (v8 ) = v7 , 3 (v7 ) = v3 , 3 (v2 ) = v5 , 4 (v2 ) = v4 . Lemma 5: Let v and w be different vectors of weight 16 in C21 . All the other vectors of weight 16 in C21 form two orbits with respect to St (v; w ) Aut (C21 ). Proof: The order of St (v; w) is (jSt (v )j=8) = 6. Without loss of generality, we can take v = v9 and w = v7 . The permutations 5 and 6 fix the vectors v and w . These two permutations generate a group H of order 6 and so St (v; w) = H . Using the equalities 5 (v1 ) = v3 , 5 (v3 ) = v2 , 6 (v1 ) = v6 , 6 (v2 ) = v5 , 6 (v3 ) = v8 , 5 (v4 ) = 6 (v4 ) = v4 , we prove that the vectors fv1 ; 1 1 1 ; v6 ; v8 g form the orbits fv1 ; v2 ; v3 ; v5 ; v6 ; v8 g and fv4 g with respect to H . Actually, the weights of the vectors v7 + v9 + v4 and v7 + v9 + vi ; i = 1; 2; 3; 5; 6; 8 are 8 and 12, respectively. These three lemmas give us that up to equivalence there are two possibilities for the generator matrix with these first four rows and the generator matrix of Res (C69 ; u1 ). These two possibilities are the matrices G069 and G0069 , respectively (see the bottom of the next page). Thus we have to check all the possibilities for the first 48 coordinates of the last five rows of G69 . We will consider only nonequivalent codes with respect to permutations of the form 1 2 where 1 is a permutation of the first 48 coordinates and 2 Aut (C21 ): By exhaustive search with the indicated restrictions we obtain the trees of extension. We obtain that there are [69; 8; 32] codes, but there are no [69; 9; 32] codes. By a dual transform the obtained [69; 8; 32] codes correspond to nonprojective optimal [84; 8; 40] codes. We also use the fact that there is only one possibility for the residual code Res (C69 ; vi ); i = 1; 2; 3; 4. An exhaustive search was done by a backtrack algorithm realized in the program EXTENSION. The running time was about 15 min on a Pentium 166 –MHz processor. More details can be found in [2]. There exists a code with parameters [70; 9; 32] (see Section IV). Thus the following is true.
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1055
1101110001000000111010100101011101011111110100101000010001010010100001000001 0011001111100001101000010001000011101001010010000001001011101001001110101111 1000100100010111000110000000110111010100100000111000011000010111110101011011 0000000011001100101001101011110000001011111110001000000111001111001011000101 0
G76
=
0111100000000011100000011000001111000111100001100100111111000000100100111111 0000011111000000011000000111111111000000010000011100000000111111100011111101 0000000000111111111000000000000000111111110000000011111111111111100000000011 0000000000000000000111111111111111111111110000000000000000000000011111111111 0000000000000000000000000000000000000000001111111111111111111111111111111111
Theorem 2:
n(9;
32) = 70.
Corollary 1: n(10; 80) > 165. Proof: Jaffe [11] proved that a putative [165; 10; 80]-code C165 has a unique weight distribution: A0 = 1; A80 = 858; A96 = 165: Res (C165 ; 96) is a code of parameters [69; 9; 32].
parameters [256; 9; 128] and a suitable kind of generator matrix of a code with parameters [51; 8; 24] in the following way: 11
G
=
111
G1
III. EXISTENCE RESULTS Theorem 3: n(9; 34) 76. Proof: Using the method from [3] we construct a [76; 9; 34] code with generator matrix G76 (see the top of this page) and weight distribution A0 = 1, A34 = 120, A36 = 140, A38 = 90, A40 = 80, A42 = 16, A44 = 8, A46 = 26, A48 = 27, and A50 = 4: Theorem 4: n(9; 146) = 297 and n(9; 148) = 300. Proof: We refer to Helleseth [12] that n(9; 146) 297 and n(9; 148) 300. Now we will construct codes of parameters [297; 9; 146] and [300; 9; 148] as follows. From Bhandari and Garg [6] we know that a code of parameters [51; 8; 24] has A24 = 204, A32 = 51, and B3 = 17. So without loss of generality we can arrange a suitable kind of generator matrix of the Reed–Muller code with
1111
000
0000
000
111
111
0000
000
1100
110
1010
101
111
G2
11 0
G
111
1
0
111
G1
G2
is a generator matrix of a code with parameters [300; 9; 148] because we lost seven columns of G and part of the corresponding rows with maximum weight 4. It is possible that after deleting three linearly dependent columns from [51; 8; 24] we obtain a code of length 48 with . B3 6= 0. Hence, we conclude that a [297; 9; 146]-code exists
000000000000111111111111111111111111111111110000011001101110111111111 000011111111000000001111111111110000111111111111111111110110110001111 =
111111111111111111111111111111111111111111111111111111
000101110000001011010
111111111111111111111111111111111111111111111111111111
001111000000001101100
111111111111111111111111111111111111111111111111111111
110000001111110000000
111111111111111111111111111111111111111111111111111111
001100001111001100000
111111111111111111111111111111111111111111111111111111
001110100100011010000
111111111111111111111111111111111111111111111111000000000000000000000 111111111111111111111111111111110000000000000000100110011110111111111 000000000000111111111111111111111111111111110000011001101110111111111 000111111111000000000111111111110001111111110111110011111001111101111 00
G69
=
00
=
111111111111111111111111111111110000000000000000100110011110111111111
0
:
A code with a generator matrix G has minimum distance 152. Then
111111111111111111111111111111111111111111111111000000000000000000000
G69
00
111111111111111111111111111111111111111111111111111111
001110100100011010000
111111111111111111111111111111111111111111111111111111
111111001111111101100
111111111111111111111111111111111111111111111111111111
000001011101011011111
111111111111111111111111111111111111111111111111111111
110101111111111011010
111111111111111111111111111111111111111111111111111111
111111110110110001111
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Remark: Codes with parameters [300; 9; 148] and [297; 9; 146] were constructed independently by Jaffe. IV. SOME CODES OF MINIMUM DISTANCE 32 AND SOME RELATED CODES
[8] S. M. Dodunekov, S. Guritman, and J. Simonis, “New results on optimal binary linear codes of dimension 9,” IEEE Trans. Inform. Theory, submitted for publication. [9] J. H. Griesmer, “A bound for error-correcting codes,” IBM J. Res. Develop., vol. 4, pp. 532–542, 1960. [10] S. Guritman, “Some new exact values and lower bounds on the minimum length of binary linear codes of dimension nine,” Fac. Inform. Technol. Syst.—Delft Univ. Technol., Delft, The Netherlands, Internal Rep. 28, 1998. [11] D. B. Jaffe. (1997, Apr.) Binary linear codes: new results on nonexistence. Dept. Math. Stat., Univ. Nebraska, Lincoln, NE. [Online] Draft version accessible through the author’s web page http://www.math.unl. edu/ djaffe, Version 0.4 [12] T. Helleseth, “A characterization of codes meeting the Griesmer bound,” Inform. Contr., vol. 50, pp. 128–159, 1981. [13] H. C. A. van Tilborg, “The smallest length of binary 7-dimensional linear codes with prescribed minimum distance,” Discr. Math., vol. 33, pp. 197–207, 1981.
Let G1 be the trivial generator matrix of the cyclic [64; 10; 28] code with generator polynomial
1111011010110 111000000 100100110011101001100011001000110000000000: Then G = G1 jG2 generates a [71; 10; 32] code of distribution A1 = 0, A32 = 574, A40 = 448, A64 = 1 where G2 is the matrix 10001111 10110011 11010101 11011010 11101001 : 10111100 11100110 10001111 10110011 11010101 A code with parameters [71; 10; 32] is known [4]. It is easy to find a nine-dimension subcode of full length 70, which does not contain a codeword of weight 64. This [70; 9; 32] code has the weight distribution A0 = 1, A32 = 315, A40 = 196. A code with such parameters are known (private communication with Brouwer). It was constructed by using a [10; 3; 8] code over a field of eight elements with nonzero codewords of weight 8 and 10. Using the codewords of weight 40 and the dual transform (see [5]) we obtain an optimal [196; 9; 96] code.
On Cyclic Reversible Self-Dual Additive Codes with Odd Length Over Moshe Ran, Senior Member, IEEE, and Jakov Snyders, Senior Member, IEEE Abstract—Several additive codes of odd length over are introduced. These codes are cyclic and reversible. Furthermore, they are self-dual under an appropriately selected binary-valued inner product. Some binary derivatives of these codes have good parameters. All cyclic and reversible [5 2 5 3] additive codes over are isomorphic and possess interesting properties. Index Terms—Additive code, cyclic code, group code, pentacode, reversible code, self-dual code.
ACKNOWLEDGMENT The authors wish to thank Prof. S. Kapralov and Dr. S. Bouyuklieva for their useful remarks and help in finding the automorphism group of the [21; 8; 8] code. This correspondence was completed during a visit of S. Guritman to the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria. He would like to express his gratitude to all the colleagues in the institute for their hospitality and support. REFERENCES [1] L. D. Baumert and R. J. McEliece, “A note on the Griesmer bound,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 134–135, 1973. [2] I. Bouyukliev, D. Jaffe, and V. Vavrek, “The smallest length of eightdimensional binary linear codes with prescribed minimum distance,” preprint. [3] I. Boukliev, “A method for construction of good linear codes and its application to ternary and quaternary codes,” in Proc. OCRT, Sozopol, Bulgaria, May 26–June 1 1995, pp. 15–20. [4] A. E. Brouwer, “Bounds on the size of linear codes,” in Handbook of Coding Theory, V. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998. [5] A. E. Brouwer and M. van Eupen, “The correspondence between projective codes and 2-weight codes,” Des., Codes Cryptogr., vol. 3, no. 11, pp. 261–266, 1997. [6] M. C. Bhandari and M. S. Garg, “A new lower bound on the minimal length of a binary linear code,” Europ. J. Combin., vol. 17, no. 4, pp. 335–342, 1996. [7] S. M. Dodunekov, S. B. Encheva, and A. N. Ivanov, “New bounds for the minimal length of binary linear block codes” (in Russian, translated from Probl. Pered. Inform., vol. 29, no. 2, pp. 41–47, 1993), Probl. Inform. Transm., vol. 29, no. 2, pp. 132–139, 1993.
I. INTRODUCTION Let G be a finite Abelian group. An additive code of length N over G is a nonempty subgroup of the N -fold direct (Cartesian) product G N . This terminology is due to Delsarte [6]. Additive codes have also been called Abelian group codes, group codes as well as block group codes, and are clearly distinguished from the group codes of Slepian [20]. An additive code C of length N , log size K = logjGj jC j, and minimum Hamming distance DH is called an [N; K; DH ] code. A linear code C over a finite field GF (q ) is also an additive code over the additive group of GF (q ). It is known that the additive group of any finite field GF (q ) is elementary Abelian [18], i.e., the order of all nonzero elements is a prime p, namely, the characteristic of GF (q ). Furthermore, it is shown in [8] that among the additive codes over Abelian groups that possess information sets, those over elementary Abelian groups have the best parameters for a given group size. Many of the best known codes over finite fields are self-dual codes. Some classes of relatively short self-dual codes have been completely Manuscript received October 22, 1998; revised August 2, 1999. This work was supported in part by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities. M. Ran is with TelesciCOM Ltd., Holon 58810, Israel (e-mail: Mran@ telesciCOM. co.il). J. Snyders is with the Department of Electrical Engineering–Systems, Tel-Aviv University, Ramat-Aviv 69978, Israel (e-mail:
[email protected]). Communicated by A. M. Barg, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(00)03097-2.
0018–9448/00$10.00 © 2000 IEEE
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