Some New Optimal Ternary Linear Codes

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1 Designs, Codes and Cryptography, 12, 5 11 (1997) c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Some New Optimal Ternary Linear Codes ILIYA BOUKLIEV* Institute of Mathematics, Bulgarian Academy of Sciences, P.O.Box 323, 5000 V. Tarnovo, Bulgaria Communicated by: H. van Tilborg Received February 23, 1996; Revised August 1, 1996; Accepted September 9, 1996 Abstract. Let d 3 (n, k) be the maximum possible minimum Hamming distance of a ternary [n, k, d; 3]-code for given values of n and k. It is proved that d 3 (44, 6) = 27, d 3 (76, 6) = 48, d 3 (94, 6) = 60, d 3 (124, 6) = 81, d 3 (130, 6) = 84, d 3 (134, 6) = 87, d 3 (138, 6) = 90, d 3 (148, 6) = 96, d 3 (152, 6) = 99, d 3 (156, 6) = 102, d 3 (164, 6) = 108, d 3 (170, 6) = 111, d 3 (179, 6) = 117, d 3 (188, 6) = 123, d 3 (206, 6) = 135, d 3 (211, 6) = 138, d 3 (224, 6) = 147, d 3 (228, 6) = 150, d 3 (236, 6) = 156, d 3 (31, 7) = 17 and d 3 (33, 7) = 18. These results are obtained by a descent method for designing good linear codes. Keywords: ternary codes, optimal codes, construction method 1. Introduction Let Fq n be the n-dimensional vector space over the Galois field F q. The Hamming distance between two vectors of Fq n is defined to be the number of coordinates in which they differ. A q-ary linear [n, k, d; q]-code is a k-dimensional linear subspace of Fq n with minimum distance d. A central problem in coding theory is to optimize one of the parameters n, k, d for given values of the other two. Two versions are: Problem 1. Find d q (n, k), the largest value of d for which there exists an [n, k, d; q]-code. Problem 2. Find n q (k, d), the smallest value of n for which there exists an [n, k, d; q]-code. A code which achieves one of these two values is called optimal. Problem 2 for the case of ternary (q = 3) codes has been considered by Hill and Newton [26]. The problem of finding n 3 (k, d) was solved for k 4 for all d, and values of n 3 (5, d) were determined for all but 30 values of d. The works of van Eupen [6], [4], van Eupen and Hill [8], van Eupen, Hamada and Watamori [7], Hamada and Watamori [21], Hamada, Helleseth and Ytrehus [20], Bogdanova and Boukliev [1], and Landgev [14] solve the problem of finding n 3 (k, d) for k = 5 for all d. The problem of finding n 3 (6, d) has been studied by many authors. Results have been obtained by Brouwer and van Eupen [2], Daskalov [3], van Eupen [6], [5], Gulliver [10], * This work was partially supported by the Bulgarian National Science Fund under Grant No. MM - 502/1995. This work was presented in part at the International Workshop OCRT 95, Sozopol, Bulgaria, May 26 June 1, 1995 and at the IEEE International Simposium on Information Theory, Whistler, Canada, September 17 22, 1995.

2 6 BOUKLIEV Gulliver and Bhargava [11], [12], Kschischang and Pasupathy [13], Hamada [15], [16], Hamada and Helleseth [17], [18], [19], Hamada and Watamori [22], Hill [24] and Hill and Greenough [25]. This paper considers ternary codes. In Section 2 a method for construction of good linear codes is presented. Section 3 includes the new results. 2. Construction Method Let S k,q be the set of all column vectors a = (a 1, a 2,...,a k ) t Fq k such that either a 1 = 1 or a 1 = a 2 = =a i 1 = 0, a i = 1 for some integer i in {2, 3,...,k}, where k 3 and Fq k denotes a k-dimensional vector space over F q. Then S k,q consists of all (q k 1)/(q 1) nonzero vectors in Fq k and the vectors in S k,q can be regarded as (q k 1)/(q 1) points in a finite projective geometry PG(k 1,q). A linear code is called projective if no two columns of a generator matrix are linearly dependent. We regard the columns of the generator matrix of a projective code as a subset of S k,q.for any n-element subset of S k,q there is a projective [n, k 0, d; q]-code, k 0 k with a generator matrix containing these elements as columns. Denote by T (C) the set of the columns of the generator matrix of the code C, considered as elements of S k,q. The Griesmer bound [9], [28] provides an important lower bound on n q (k, d): k 1 n q (k, d) g q (k, d) = d/q i. i=0 LEMMA 1 ([23]) Suppose that d q k 1 andthatcisan[n,k,d;q]-code which attains the Griesmer bound. Then C is projective. The problem of the construction of a [g q (k, d), k, d; q] code (if such a code exists) can be formulated in the following way: Find n elements of S k,q such that the [n, k, d; q] code meets the Griesmer bound. We use the following notations: 1. Each n-element subset of S k,q is called a solution. 2. The substitution of a given element of a solution s by another one is called an elementary transformation. 3. The neighbourhood N(s) of a solution s is the set of all solutions s that can be obtained from s by an elementary transformation. 4. We define the evaluation function by f (s) = d min (s)q k + q k A dmin (s), where d min (s) is the minimum distance of the corresponding code and A dmin (s) is the number of codewords of minimum weight in the corresponding code.

3 SOME NEW OPTIMAL TERNARY LINEAR CODES 7 The problem of the construction of a [g q (k, d), k, d; q] code (if such one exists) if considered as a combinatorial optimization problem is the following max{ f (s), s S k,q, s =n}. The construction method is based on the next heuristic algorithm: variables s, s : solution; Number of Descents:integer; begin Number of Descents:=Random(1000); while Number of Descents>0 do begin {descent } Get Initial Solution (s); while exists s in N(s) such that f(s ) > f(s) do s:=s ; if d min (s) = d target then Output Solution (s); Number of Descents:=Number of Descents 1; end; {descent } end. The initial solution s can be chosen in several ways: (i) By using a code C with parameters [n l, k, d δ; q] where δ l. In this case l elements from S k,q, not belonging to T (C ) are randomly added to T (C ). (ii) By using a code C with parameters [n + l, k, d + l 1; q]. In this case l elements of T (C ) are randomly deleted. This algorithm can be used for constructing optimal codes not meeting the Griesmer bound. If d > q k 1 the set S k,q must be taken several times. For all codes constructed in the paper by this method the set S k,q is taken twice. Some examples are given in the next table: Initial Solution Result [120, 6, 78; 3] [124, 6, 81; 3] [124, 6, 81; 3] [130, 6, 84; 3] [160, 6, 105; 3] [156, 6, 102; 3] [156, 6, 102; 3] [152, 6, 99; 3] Other versions of the descent method are used in [11], [27]. 3. The New Results The results in dimension 6 are summarized in Theorem 1.

4 8 BOUKLIEV THEOREM 1 There exist codes with parameters (i) [44, 6, 27; 3], [76, 6, 48; 3], [94, 6, 60; 3], [124, 6, 81; 3], [130, 6, 84; 3], [134, 6, 87; 3], [138, 6, 90; 3], [148, 6, 96; 3], [152, 6, 99; 3], [156, 6, 102; 3], [164, 6, 108; 3], [170, 6, 111; 3], [179, 6, 117; 3], [188, 6, 123; 3],[206, 6, 135; 3] [211, 6, 138; 3], [224, 6, 147; 3], and [236, 6, 156; 3]; (ii) [228, 6, 150; 3]. Proof. (i) Codes with such parameters are constructed by the method from Section 2. Theirweight enumerators are: [44, 6, 27; 3] z z z z 36, [76, 6, 48; 3] z z z z 57, [94, 6, 60; 3] z z z z 72, [124, 6, 81; 3] z z z 99, [130, 6, 84; 3] z z z z z z z 105, [134, 6, 87; 3] z z z z z 105, [138, 6, 90; 3] z z z z 117, [148, 6, 96; 3] z z z z z 114, [152, 6, 99; 3] z z z z z 117, [156, 6, 102; 3] z z z z z 120, [164, 6, 108; 3] z z z 126, [170, 6, 111; 3] z z z z z z z 132, [179, 6, 117; 3] z z z z z z z z 141, [188, 6, 123; 3] z z z z z 141, [206, 6, 135; 3] z z z 153, [211, 6, 138; 3] z z z z z 162, [224, 6, 147; 3] z z z z 171 [236, 6, 156; 3] z z z z 183. The description of the corresponding generator matrices is given in Appendix. (ii) This code has been constructed as a quasi-twisted one with a generator matrix consisting of 38 twistulant matrices with first rows , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , The weight enumerator of the code is z z z z 174. For quasi-twisted codes see [25]. All codes described above are optimal with respect to minimum distance (See [22]). THEOREM 2 d 3 (31, 7) = 17 and d 3 (33, 7) = 18. Proof. From [22] we have d 3 (31, 7) 17 and d 3 (33, 7) 18.

5 SOME NEW OPTIMAL TERNARY LINEAR CODES 9 Codes with parameters [31, 7, 17; 3] and [33, 7, 18; 3] were constructed by the method from Section 2. Their weight enumerators and generator matrices are [31, 7, 17; 3] 1+324z z z z z z z z z 25 +2z 27, [33, 7, 18; 3] 1+348z z z z z z z z z z 27, Appendix. Generator Matrices of Codes from Theorem 1 Let the vectors from S k,q be arranged in lexicographic order. A generator matrix of a projective code is represented by a binary vector of length (q k 1)/(q 1) with 1 indicating the presence, and 0 the absence of a vector of S k,q. This binary vector is then broken into blocks of length 4, each of which is represented by a hexadecimal symbol from {0, 1,...,9,a,b,c,d,e, f}. If it is necessary the last block is beforehand completed by zeroes. Example. if q = 3 and k = 3 the set S k,q looks like that: Then the code [8,3,4;3] with generator matrix:

6 10 BOUKLIEV is represented in the following way: (2) = e1d8 (16). [44, 6, 27; 3] c a a a008; [76, 6, 48; 3] 1420a c80c bc c00100; [94, 6, 60; 3] d200842d b84c451e e310 03a ; [124, 6, 81; 3] 1660b c c28342d1089dec0132d20c f213c c025e011 0d002133a19; [129, 6, 83; 3] 1662b c c29342d1589dec0132d20c c0906f213c e011 0d002133a19; [134, 6, 87; 3] 1661b c d039c29342d1589dec0932d20c c0906f213c c025e011 0d002133a19; [138, 6, 90; 3] e044040c0131bdea210f0dc bead20051c8cac7e1736bfd4c47018a03108be aa551005; [148, 6, 96; 3] dead645c41e21901a06a2be29d29216b ae100b1a39a24a98dc560450b34a a8a5 710ad281951; [152, 6, 99; 3] b5b2c052d9d b06672d02da52722b95b05114e8a d369144b26a281f80b b2222b94; [156, 6, 102; 3] b5b2c0d2d9d b06672d02da52722b95b05114e8a d369144b26a281f82b b2222b95; [164, 6, 108; 3] b5b2c0d2d9d c2b06672d02da52722b95b05134e8e d369144b26a283f82b b2223b95; [170, 6, 111; 3] c356044b3608e7ea401fdc68fc220b81aec773b b646f90976b1f70bcd16826efb ad2248d2bd8; [178, 6, 116; 3] 02dfe85c357684b364ae7ea401f9c68ec220a018ec773b b646f90976b1f709cd16826efb ad2268d2bdc; [188, 6, 123; 3] 23c00946aaa542cf23c2917cf5671ea541c19e721db71faf34805eee30b5db22c73add2d373acb4d d6bbad7cc47; [206, 6, 135; 3] 35f8e77d3d485b5bde09ffab016fe33ed74d9c3c761a2f13caffd5024f437df4dfe5141f96d26fbb 54f65f18121; [210, 6, 137; 3] 29d1265f7c4335e369467dae2dedf1ae49c7b9f8b1ee8e1b36d36c96b5861beefb60f6dde597e7f4 d1b6bfae12d; [224, 6, 147; 3] fba476f5c3cd0be3bced71eeead37a785ea734de5ed9ca37e7adacedd7ca3f6b6add57b1ddd71cee f8728fad3d3; [12, 6, 6; 3] the extended ternary Golay code, a0000; The generator matrices of the [130,6,84;3], [179,6,117;3] and [211,6,138;3] codes are obtained by adding the vectors (0,1,0,1,2,2) t, (0,1,0,0,0,1) t and (1,2,1,2,1,2) t respectively to the generator matrices of the [129,6,83;3], [178,6,116;3] and [210,6,137;3] codes. We obtain a generator matrix of a non-projective [236,6,156;3] code concatenating the generator matrices of the [12,6,6;3] and [224,6,147;3] codes given above. Acknowledgments The author thanks Prof. S. Dodunekov, Prof. S. Kapralov and Dr. R. Daskalov for their helpful remarks.

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