Báo cáo toán học: "New Optimal Constant Weight Codes" pot

published a new table of constant weight codes, updating existing tables originally created by Brouwer et al.. This paper improves upon these results by filling in 9 missing constant wei

New Optimal Constant Weight Codes

I Gashkov∗, D Taub

Department of Mathematics Karlstad University, Sweden igor.gachkov@kau.se,taub.math@gmail.com Submitted: May 18, 2007; Accepted: Jun 18, 2007; Published: Jun 21, 2007

Mathematics Subject Classifications: 94B60

Abstract

In 2006, Smith et al published a new table of constant weight codes, updating existing tables originally created by Brouwer et al This paper improves upon these results by filling in 9 missing constant weight codes, all of which are optimal by the second Johnson bound This completes the tables for A(n, 16, 9) and A(n, 18, 10)

up to n = 63 and corrects some A(n, 14, 8)

Introduction

A binary constant weight code is any subset of n

such that all elements, codewords, have the same weight An important problem in coding theory is finding A(n, d, w), the maximum possible number of codewords in a constant weight code with length n, minimum distance d, and weight w

A large table of lower bounds on these numbers was published by Brouwer et al [1], and later added to by Smith et al [4] However, these tables are still far from complete, and many of the existing values can be improved upon

When the lower bound for a code matches the upper bound, the code is said to be optimal There are a number of different methods for determining upper bounds for constant weight codes, but for the codes presented in this paper the most useful is the one given by theorem 1

Theorem 1 (The second Johnson bound) There can be a code with parameters n, d and w and size M only if

(n − b)a(a − 1) + ba(a + 1) ≤ w−

&

d 2

'!

M(M − 1) holds, where a and b are the unique integers such that wM = an + b and 0 ≤ b < n

Proof See [2], p 526.

The second Johnson bound J2(n, d, w) is the largest M such that the inequality in theorem 1 holds (It is possible that the inequality holds for all M , in which case

J2(n, d, w) = ∞.)

All of the codes presented in this paper are optimal by the second Johnson bound

New Lexicographic Methods

In general, lexicographic methods, or lexicodes, rarely achieve good lower bounds How-ever, with some simple modifications, standard lexicodes can be used to obtain a number

of useful results

By adding a degree of randomness to a standard lexicode we were able to obtain a number of new optimal constant weight codes Additional codes were found by using a genetic algorithm based on randomized lexicodes These new codes complete the tables for A(n, 16, 9) and A(n, 18, 10) up to n = 63

A complete list of all new lower bounds, as well as the actual codes, is presented below

We presented binary vector v = (b1, b2, , bn) in support form {i | bi = 1}

Results

The results are given in two tables All codes presented in this section are optimal by the second Johnson bound

Table 1

n: 39 40 41 42 43 44 45

A(n,14,8): 10 10 11 12 12 13 14

A(39, 14, 8) = 10,A(41, 14, 8) = 11,A(42, 14, 8) = 12 (See [6])

A(40, 14, 8) = 10 the same as A(39, 14, 8)

A(42, 14, 8) = 12 the same as A(41, 14, 8)

A(45, 14, 8) = 14

Table 2

n: 45 46 47 48 48 50 51 52 53 54 55 56 57 58 59 60 61 62 63

A(n,16,9): 10 10 10 11 11 12 12 13 13 14 15 16 19 19 20 21 22 24 28

A(45, 16, 9) = 10 (See [3])

A(46, 16, 9) = 10 and

A(47, 16, 9) = 10 the same as A(45, 16, 9)

A(48, 16, 9) = 11 (See [5], p 912-915.)

A(49, 16, 9) = 11 the same as A(48, 16, 9)

A(50, 16, 9) = 12

A(51, 16, 9) = 12 the same as A(50, 16, 9) See also [5], p 912-915

A(52, 16, 9) = 13

A(53, 16, 9) = 13 the same as A(52, 16, 9)

A(54, 16, 9) = 14

A(55, 16, 9) = 15

A(56, 16, 9) = 16 (See [3])

A(57, 16, 9) = 19 (See [3])

A(58, 16, 9) = 19 the same as A(57, 16, 9)

A(59, 16, 9) = 20

A(60, 16, 9) = 21

A(61, 16, 9) = 22

A(62, 16, 9) = 24 and A(63, 16, 9) = 28 (See [3])

Table 2

n: 55 56 57 58 59 60 61 62 63

A(n,18,10): 11 11 11 12 12 12 13 13 14

A(55, 18, 10) = 11 (See [3])

A(56, 18, 10) = 11 and

A(57, 18, 10) = 11 the same as A(55, 18, 10)

A(58, 18, 10) = 12 (See [5], p 912-915.)

A(59, 18, 10) = 12 and

A(60, 18, 10) = 12 the same as A(58, 18, 10)

A(61, 18, 10) = 13

A(62, 18, 10) = 13 the same as A(61, 18, 10)

A(63, 18, 10) = 14

References

[1] A E Brouwer, J B Shearer, N J A Sloane, W D Smith, “A New Table of Constant Weight Codes,” IEEE Trans Inform Theory 36 (1990)

[2] F J MacWilliams, N J A Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1979)

[3] E M Rains, N J A Sloane, “Table of Constant Weight Binary Codes,” http://www.research.att.com/~njas/codes/Andw/

[4] D H Smith, L A Hughes and S Perkins, “A New Table of Constant Weight Codes

of Length Greater than 28,” Electron J Combin 13 (2006)

[5] I Gashkov,“Optimal Constant Weight Codes,” Lecture note in Computer science, LNCS 3991 (2006)

[6] I Gashkov, J.Ekberg, D.Taub,“A Geometric Approach to Finding New Lower Bounds

of A (n, d, w),” Designs, Codes and Cryptography 43:2/3 June 2007