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The Z_4-linearity 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 Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the ..."
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Cited by 178 (15 self)
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Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. 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 Nordstrom-Robinson code is self-dual — which explains why their weight distributions are dual to each other. The Kerdock and ‘Preparata ’ codes are ¡ 4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over ¡ 4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the ‘Preparata ’ code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over ¡ 4, but extended Hamming codes of length n ≥ 32 and the
Duality for modules over finite rings and applications to coding theory
- AMER. J. MATH
, 1999
"... This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite ..."
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Cited by 58 (5 self)
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This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters.
The Nordstrom-Robinson code is the binary image of the octacode
- Proceedings DIMACS/IEEE Workshop on Coding and
, 1992
"... The Nordstrom-Robinson code, a nonlinear binary code of length 16 and minimal Hamming distance 6, is the binary image of the octacode, a linear self-dual code over 4 of length 8 and minimal Lee distance 6. Since the octacode is the 4-analogue of a Hamming code, this ..."
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Cited by 29 (7 self)
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The Nordstrom-Robinson code, a nonlinear binary code of length 16 and minimal Hamming distance 6, is the binary image of the octacode, a linear self-dual code over 4 of length 8 and minimal Lee distance 6. Since the octacode is the 4-analogue of a Hamming code, this
Weight Functions and the Extension Theorem for Linear Codes over Finite Rings
, 1999
"... An extension theorem for general weight functions is proved over finite chain rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. ..."
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Cited by 9 (3 self)
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An extension theorem for general weight functions is proved over finite chain rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof.
Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others
, 1994
"... Certain nonlinear binary single-error-correcting codes found by Julin, Best and others have simple descriptions as codes over the ring of integers modulo 4. ..."
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Cited by 8 (3 self)
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Certain nonlinear binary single-error-correcting codes found by Julin, Best and others have simple descriptions as codes over the ring of integers modulo 4.
On Minimal Realization Over a Finite Chain Ring.
, 2001
"... Let R be a finite chain ring, e.g. a Galois ring. We give a compact recursive formula for a minimal realization of a finite R-sequence. In particular, we show how to obtain a monic minimal polynomial and a rational approximation of a finite R-sequence. We also show how to solve the classical key equ ..."
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Cited by 3 (0 self)
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Let R be a finite chain ring, e.g. a Galois ring. We give a compact recursive formula for a minimal realization of a finite R-sequence. In particular, we show how to obtain a monic minimal polynomial and a rational approximation of a finite R-sequence. We also show how to solve the classical key equation of Algebraic Coding Theory over R.
New bounds on a hypercube coloring problem and linear codes
, 2000
"... In studying the scalability of optical networks, one problem arising involves coloring the vertices of the n-dimensional hypercube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of O/_k(n), the minimu ..."
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Cited by 2 (0 self)
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In studying the scalability of optical networks, one problem arising involves coloring the vertices of the n-dimensional hypercube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of O/_k(n), the minimum number of colors needed, appears tobe a difficult problem. In this paper, we improve the known An n-cube (or n-dimensional hypercube) is a graph whose vertices are the vectors of the
Semigroup Rings And The Extension Theorem For Linear Codes
- the Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control, and Computing
, 1997
"... . An extension theorem for general weight functions is proved over finite commutative local principal ideal rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. 1. Background In her doctoral dissertation, MacW ..."
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. An extension theorem for general weight functions is proved over finite commutative local principal ideal rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. 1. Background In her doctoral dissertation, MacWilliams [8], [9] proved an equivalence theorem: two linear codes C 1 ; C 2 ae F n defined over a finite field F are equivalent up to monomial transformations if and only if there is a linear isomorphism f : C 1 ! C 2 which preserves Hamming weight. Bogart et al. [2] gave another proof of this theorem, and a character theoretic proof was provided by Ward and the author [13]. Following up on the ideas in [13], the author has extended the character theoretic techniques to linear codes defined over finite Frobenius rings, first for the Hamming weight [15] and then for symmetrized weight compositions [16]. In this paper, the author treats general weight functions defined over finite commutat...
Network coding with modular lattices
- Journal of Algebra and Its Applications
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