On the Preparata and Goethals codes

Baker, Rodney C.; Lint, JH van; Wilson, R.

doi:10.1109/tit.1983.1056675

Cited by 61 publications

(97 citation statements)

References 6 publications

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“…3 = Best ([12], [45], p. 140). 4 = From 5 (5,6,12), 6 disjoint words of weight 2 and complements ( [45], p. 139, [132], p. 585). 5 = Romanov [155]-see Section VI.…”

Section: S5mentioning

confidence: 99%

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A new table of constant weight codes

Brouwer

Shearer

Sloane

et al. 1990

IEEE Trans. Inform. Theory

359

276

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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

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“…3 = Best ([12], [45], p. 140). 4 = From 5 (5,6,12), 6 disjoint words of weight 2 and complements ( [45], p. 139, [132], p. 585). 5 = Romanov [155]-see Section VI.…”

Section: S5mentioning

confidence: 99%

“…5 = Romanov [155]-see Section VI. 6 = From Hamming code over GF (5) [79]. 7 = From the ulu + u construction ([132], p. 76, [166]).…”

Section: S5mentioning

confidence: 99%

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A new table of constant weight codes

Brouwer

Shearer

Sloane

et al. 1990

IEEE Trans. Inform. Theory

359

276

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“…The first construction of such codes of any admissible length n was proposed by Preparata in 1968 (see [10]). Using special automorphisms of the Galois field GF (n/2), the series of pairwise nonequivalent Preparata codes of any admissible length n was constructed by Dumer [11] and later by Baker, van Lint, and Wilson [12]. Preparata codes are exceptional in some prominent properties.…”

Section: Distance Regularity and Preparata Codesmentioning

confidence: 99%

On distance nonregularity of Preparata codes

Solov’eva

Tokareva

2007

Sib Math J

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“…The main unsolved question concerning these codes has always been whether they are duals in some more algebraic sense. Many authors have investigated these codes and have found that (except for the Nordstrom-Robinson code) they are not unique and, indeed, that large numbers of codes exist with the same weight distributions [BLW83,Ca89,Ka82,Ka82a,Ka83,VL83]. Kantor [Ka83] declares that the "apparent relationship between these [families of codes] is merely a coincidence".…”

Section: Introductionmentioning

confidence: 99%

A linear construction for certain Kerdock and Preparata codes

Calderbank¹,

Hammons²,

Kumar³

et al. 1993

Bull. Amer. Math. Soc.

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Abstract.The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over Z4 , the integers mod 4 . The Kerdock and Preparata codes are duals over Z4 , and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over Z4 . This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First-and second-order Reed-Muller codes are also linear codes over Z4 , but Hamming codes in general are not, nor is the Golay code.

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On the Preparata and Goethals codes

Cited by 61 publications

References 6 publications

A new table of constant weight codes

A new table of constant weight codes

On distance nonregularity of Preparata codes

A linear construction for certain Kerdock and Preparata codes

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