Four fundamental parameters of a code and their combinatorial significance
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TLDR
A method is suggested to calculate the weight distributions of cosets of a code and a “dual concept” of that of perfect codes is presented and examined in detail.Abstract:
Starting from the distance distribution of an unrestricted code and its Mac Williams transform, one defines four parameters that, in the linear case, reduce to the minimum weight and the number of distinct weights of the given code and of its dual. In the general case, one exhibits the combinatorial meaning of these parameters and, using them, one obtains various results on the distance properties of the code. In particular, a method is suggested to calculate the weight distributions of cosets of a code. A “dual concept” of that of perfect codes is also presented and examined in detail.read more
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The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes
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The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes
TL;DR: In this paper, it was shown that all the nonlinear binary codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals can be constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4.
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Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes
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TL;DR: The "coding theory" point of view for studying the existence of almost bent functions is developed, showing explicitly the links with cyclic codes and new characterizations are given by means of associated Boolean functions.
References
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Book
Algebraic Coding Theory
TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.
Journal ArticleDOI
Maximum distance q -nary codes
TL;DR: Examples and construction methods are given to show that these codes exist for a number of values of q, k, and r and with the restriction that the codes be linear.
Journal ArticleDOI
A theorem on the distribution of weights in a systematic code
TL;DR: The spectrum of a systematic code determines uniquely the spectrum of its dual code (the orthogonal vector space) and the two sets of integers are related by a system of linear equations.