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INTRODUCTION The problem of classification of optimal ternary constant-weight codes (TCW) is considered We use combinatorial and computer methods to find inequivalent codes for some cases for 3 ≤ d ≤ n ≤ 9

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PRELIMINARIES Ternary (n,M,d) code - consists of M vectors of length n over alphabet {0,1,2} such that any two codewords differ in at least d positions (n,M,d,w) constant-weight code - all the codewords have the same Hamming weight w A 3 (n,d,w) – the largest possible value of M when the other parameters are fixed Codes with such parameters are called optimal One of the main open problem of algebraic coding theory is enumeration (up to equivalence) of optimal codes

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HISTORY V. Zinoviev (1984) – Generalization of Johnson bound for constant- weight code A.Brouwer, J.Shearer, N.Sloane, and W.Smith (1990) - Table of binary constant-weight codes for n ≤ 28 E.Agrell, A.Vardy, and K.Zeger (2000) – Upper Bounds, Binary codes G.Bogdanova (2000) – New bounds, TCW codes P.Ostergard and M.Svanstrom (2002) - Table of TCW codes D. Smith, L.Hughes and S.Perkins (2006) - New Table of Constant Weight Codes for n ≥ 28 Y.Chee, S.Ling (2007) - Constructions for q-ary Constant-Weight Codes

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PRELIMINARIES A q (n,d,n) = A q-1 (n,d) Johnson bounds: −

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ENUMERATION OF TCW CODES Two ternary constant-weight codes are equivalent if one of them can be obtained from the other by transformations of the following types: – permutation of the coordinates of the code; – permutation of the alphabet symbols appearing in a fixed position Exhaustive search is not applicable for large parameters

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ENUMERATION OF TCW Maximum Clique search problem – vertex set corresponds to the words of length n and Hamming weight w – two vertices are joined by an edge if the Hamming distance between the corresponding words is greater than or equal to d (n,M,d,w) code can be shortened to get (n - 1, M',d,w) subcode Construct a code C - by classifying all such subcodes, and then use the clique-finding approach to find the rest of the words in C.

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ENUMERATION OF TCW Two basic steps: – Finding all inequivalent possibilities for subcode C 0 – Extending any of them to the size of C Аlgorithms are implemented in the computer package QPlus Some of the results are also verified using Q-Extension software

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RESULTS Theorem (a) There exist unique (up to equivalence) TCW codes with parameters: (3,3,3,2), (4,4,3,2), (4,2,4,2), (4,8,3,3), (4,2,4,3), (5,5,3,2), (5,2,4,2), (5,12,3,3), (5,5,4,3), (5,2,5,3), (5,5,4,4), (5,2,5,4), (6,3,4,2), (6,4,5,3), (6,2,6,3), (6,15,4,4), (6,4,5,4), (6,3,6,4), 6,12,4,5), (6,3,5,5), (6,2,6,5), (7,3,4,2), (7,14,4,3), (7,2,6,3), (7,2,7,4), (7,2,7,5), (7,7,5,6), (7,2,6,6), (7,2,7,6), (8,4,4,2), (8,5,5,3), (8,2,6,3), (8,2,8,4), (8,2,8,5), (8,2,8,6), (8,16,5,7), (8,2,8,7), (9,4,4,2), (9,3,6,3), (9,3,7,4), (9,2,8,4), (9,5,7,5), (9,3,8,5), (9,2,9,5), (9,3,9,6), (9,3,8,7), (9,2,9,7), (9,3,7,8),(9,2,9,8)

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RESULTS Let #(n,M,d,w) denote the number of inequivalent TCW codes with the specified parameters. (b) We have: #(5,10,3,4)=64, #(6,6,3,2)=2, #(6,18,3,3)=54, #(6,8,4,3)=3, #(6; 24; 3; 5) ≥ 20, #(7,7,3,2)=2, #(7,4,5,3)=2, #(7,7,5,4)=45, #(7,3,6,4)=3, #(7,3,6,5)=4, #(7,9,5,5)=2, #(7; 14; 4; 6) ≥ 74, #(8,8,3,2)=3, #(8,5,6,4)=2, #(8,2,7,4)=2, #(8,8,6,5)=5, #(8,3,7,5)=3, #(8,8,6,6)=22, #(8,3,7,6)=2, #(8,4,6,7)=2, #(8,2,7,7)=2, #(9,9,3,2)=4, #(9,6,5,3)=2, #(9,6,7,6)=12, #(9,3,8,6)=4, #(9,5,7,7)=11, #(9,2,8,8)=2

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EXAMPLE 00011111 00102222 01220012 10221200 11012020 11100101 22010202 22101010 00011111 00102222 01220012 10221200 11012020 12100110 21101001 22010202 00011111 00102222 01220012 10221200 12012002 12100110 21010220 21101001 00011111 01100122 01212200 10120201 12001220 20202021 21021002 22110010 00011111 01100122 02222100 10102201 12010220 20220021 21012002 22101010 Operation done in: 0:3:21 Result: Vector space: 830 Total of 7883 constant-weight codes found with parameters: (Q = 3; N = 8; M = 8; D = 6; W = 5;) 5 non equivalent codes found.

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THANK YOU !

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