A new table of constant weight codes

Abstract

Abstract-A table of binary constant weight codes of length n <; 28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and also a table is given of (unrestricted) binary codes of length J1 <; 28. I. I'ITRODUCTIONT HE MAIN GOAL of this paper is to givc an exten sive table of lower bounds on A(n, d, w), the maximal possible number of binary vectors of length n, Hamming distance at least d apart, and constant weight w. We also give a table of lower bounds on A(n, d), the maximal possible number of binary vectors of length n and Ham ming distance at least d apart (with no restriction on weight).These functions have been studied by many authors, and were tabulated for n:<:: 24 in [13], [45], [72], [132]. In the present paper we extend the tables to length n:<:: 28.Our main concern is with Table I, the table of constant weight codes. The majority of the 600 codes in this table are new, either because we have discovered nicer versions of existing codes, or (more frequcntly) because we have found better codes than were known before.Our goal has been to give either an explicit construc tion or a reference for every code in the table. With some exceptions a rcadcr should be able to reconstruct any of these codes from the information given here. (This is in contrast to [13], where several codes are simply describcd as being found by an unstatcd "miscellaneous construc tion".) However, because of space limitations, we have not included explicit listings for the codes constructed in Section XII (indicated by "y" in Table 1) when they contain more than 1500 codewords.Although [13] gives both upper and lower bounds on A(n, d, w) and A(n, d), in the present paper we give only lower bounds, i.e. tables of actual codes. We have not To save space we have sometimes written vectors in hexa decimal, using 0 = 0000,' . " 9 = 1001, A = 1010, . . " F = 1111, usually omitting leading zeros (so the vectors are right-justified). Superscripts (for example in Table XV) indicate the number of vectors in an orbit. Parenthe scs inside a vector (for example in Tables XII-XIV) indicate that all simultaneous cyclic shifts of the paren thesized sections are to be used. For example (110)(10) is an abbreviation for the six vectors 11010,01101,10110, 11001,01110,10101.A design (X, . 9]) is a set X (of "points") together with a collection [f8 of subscts of X (called "blocks"). A t -(V,k,A) design is a design in which IXI = v, all blocks contain exactly k points, and any t distinct points of X More generally an (r, A)-design is a design in which cach point belongs to exactly r blocks and each pair of points belong to exactly A blocks (but the blocks need not all contain the same number of points). A symmetric design

Constructions for constant-weight ICI-free codes

“…Under the assumption that all cells leak charge at the same rate, the relative levels of the cells remain unchanged. Both constant-weight codes [2] and balanced codes can thus be utilized to determine whether the read level should be adjusted; constant-weight codes must contain some constant proportion p of ones in every codeword, while balanced codes are a special case of constant-weight codes where there are an equal number zeros and ones in each codeword (p = 1/2). Thus, for a constant-weight code of length n and ratio p, if we read fewer than np ones in a codeword, we deduce that the cell levels have dropped, and we should read at a lower threshold.…”

New constructions for covering designs

“…Greedy coverings are not in general optimal, but as happens with codes (Brouwer, Shearer, Sloane, and Smith [3], Brualdi and Pless [4], Conway and Sloane [6]) they are often quite good-about 42% of the table entries come from greedy coverings. Interestingly, the Steiner system S (24,8,5), which Conway and Sloane [6, page 347] showed is a constant-weight lexicographic code, also arises as a greedy covering.…”

Asymptotic Bounds on the Equilateral Dimension of Hypercubes

“…In addition we refer e.g. to Brower et al [5], Heng, Cooke [10] or Fu et al [8] for recent results on equidistant and constant weight codes. In this article we study equilateral sets in the hypercube.…”