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The Z_4linearity of Kerdock, Preparata, Goethals, and related codes
, 2001
"... Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the ..."
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Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over ¡ 4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the ¡ 4 domain implies that the binary images have dual weight distributions. The Kerdock and ‘Preparata ’ codes are duals over ¡ 4 — and the NordstromRobinson code is selfdual — which explains why their weight distributions are dual to each other. The Kerdock and ‘Preparata ’ codes are ¡ 4analogues of firstorder ReedMuller and extended Hamming codes, respectively. All these codes are extended cyclic codes over ¡ 4, which greatly simplifies encoding and decoding. An algebraic harddecision decoding algorithm is given for the ‘Preparata ’ code and a Hadamardtransform softdecision decoding algorithm for the Kerdock code. Binary first and secondorder ReedMuller codes are also linear over ¡ 4, but extended Hamming codes of length n ≥ 32 and the
Symplectic semifield planes and Z4linear codes
 TRANSACTIONS OF THE AMERICAN MATHENATICAL SOCIETY
, 2004
"... There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z4–linear Kerdock and Preparata codes on the other. These inter– relationships lead to the construction of large numbers of objects of each typ ..."
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Cited by 15 (6 self)
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There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z4–linear Kerdock and Preparata codes on the other. These inter– relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non–isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z4–linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of “boring ” affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes “boring ” in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.
A linear construction for certain Kerdock and Preparata codes
 Bull. Amer. Math. Soc
, 1993
"... codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the NordstromRobinson code is selfdual. All these codes are just extended cyclic codes over Z4. This provides a simple definition for these codes and explains why their Hamming weight d ..."
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Cited by 15 (3 self)
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codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the NordstromRobinson code is selfdual. 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 secondorder ReedMuller codes are also linear codes over Z4, but Hamming codes in general are not, nor is the Golay code. 1.
Quaternionic linesets and quaternionic Kerdock codes
 LINEAL ALGEBRA AND ITS APPPLICATIONS
, 1995
"... When n is even, orthogonal spreads in an orthogonal vector space of type O−(2n−2,2) are used to construct linesets of size (2n−1+1)2n−2 in H2n−2 all of whose angles are 90° or cos−1(2−(n−2)/2). These linesets are then used to obtain quaternionic Kerdock codes. These constructions are based on idea ..."
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Cited by 5 (2 self)
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When n is even, orthogonal spreads in an orthogonal vector space of type O−(2n−2,2) are used to construct linesets of size (2n−1+1)2n−2 in H2n−2 all of whose angles are 90° or cos−1(2−(n−2)/2). These linesets are then used to obtain quaternionic Kerdock codes. These constructions are based on ideas used by Calderbank, Cameron, Kantor, and Seidel in real and complex spaces.
Orthogonal spreads and translation planes
"... There have hccn a nmnhcr of striking new fesults concerning translation planes of characteristic 2, ohtained using orthogonal and sYlnplcctic spreads. The iInpdus for this came from coding theory. This paper surveys the gCOlndric advances, while providing a hint of their coding theoretic connections ..."
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Cited by 3 (0 self)
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There have hccn a nmnhcr of striking new fesults concerning translation planes of characteristic 2, ohtained using orthogonal and sYlnplcctic spreads. The iInpdus for this came from coding theory. This paper surveys the gCOlndric advances, while providing a hint of their coding theoretic connections.
Orthogonal Dual Hyperovals, Symplectic Spreads and Orthogonal Spreads
"... Orthogonal spreads in orthogonal spaces of type V + (2n + 2, 2) produce large numbers of rank n dual hyperovals in orthogonal spaces of type V + (2n, 2). The construction resembles the method for obtaining symplectic spreads in V (2n, q) from orthogonal spreads in V + (2n + 2, q) when q is even. ..."
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Orthogonal spreads in orthogonal spaces of type V + (2n + 2, 2) produce large numbers of rank n dual hyperovals in orthogonal spaces of type V + (2n, 2). The construction resembles the method for obtaining symplectic spreads in V (2n, q) from orthogonal spreads in V + (2n + 2, q) when q is even.
NorthHolland Kerdock codes and related planes
, 1992
"... Among the many aspects of coding theory Jack van Lint has studied intensively are some generalizations of Preparata and Kerdock codes (see Baker et al. (1983), Cameron and Van Lint (1991) and Van Lint (1983)). There are still many open problems concerning these. This note is a brief discussion of pr ..."
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Among the many aspects of coding theory Jack van Lint has studied intensively are some generalizations of Preparata and Kerdock codes (see Baker et al. (1983), Cameron and Van Lint (1991) and Van Lint (1983)). There are still many open problems concerning these. This note is a brief discussion of problems and new results involving orthogonal spreads, translation planes and associated generalized Kerdock codes. 1. Orthogonal spreads Let V be a vector space of dimension 4m over a finite field L of characteristic 2, where m 2 2. Assume that V is equipped with a quadratic form Q of Witt index 2m; the associated bilinear form is denoted (u, u). Then the pair V, Q is equivalent to the pair L4m, Qdm, where Write the standard ordered basis of L4m as e,,..., ezm,fi,...,hrn, so that
Kerdock codes and extremal Euclidean linesets
, 802
"... Association schemes related to universally optimal configurations, ..."
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