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Self-dual codes. (English) Zbl 0936.94017

Pless, V. S. (ed.) et al., Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 177-294 (1998).
For the entire collection see Zbl 0907.94001. This chapter provides a comprehensive and up-to-date survey of results on self-dual codes. Codes over \(\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_q, \mathbb{Z}_4\) and \(\mathbb{Z}_m\) are considered. Bounds and extremal codes are given, and shadow codes, weight enumerators, mass formulae and enumeration are discussed. Tables of bounds are given as well as examples of extremal codes. The enumeration of codes of small length is also given. Weight enumerators are given and their bases are found using invariant theory. An extensive bibliography is given with 336 references.

MSC:

94B05 Linear codes (general theory)
94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory
94B65 Bounds on codes

Citations:

Zbl 0907.94001
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Online Encyclopedia of Integer Sequences:

Weight distribution of binary Golay code of length 24.
Number of self-dual codes of length 2n over GF(4).
Number of indecomposable self-dual codes of length 2n over GF(4).
Number of indecomposable self-dual binary codes of length 2n.
Number of self-dual binary codes of length 2n (up to column permutation equivalence).
Expansion of g.f.: (1+x^9)/((1-x^4)*(1-x^6)).
Highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.
Weight distribution of hypothetical [ 72,36,16 ] doubly-even binary self-dual code.
Molien series for complete weight enumerator of self-dual code over GF(4) containing 1^n.
Molien series for complex 4-dimensional Clifford group of order 92160 and genus 2. Also Molien series of ring of biweight enumerators of Type II self-dual binary codes.
Weight distribution of the dodecacode, a [[12,0,6]] quantum-error-correcting code.
Molien series for ring of symmetrized weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).
Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.
Molien series for complete weight enumerator of self-dual code over GF(5).
Molien series for complete weight enumerator of self-dual code over GF(5) containing all-1’s vector.
Weight distribution of hypothetical [ 24,12,10 ] trace-self-dual additive code over GF(4).
Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.
Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.
Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.
Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.
Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105681.
Number of codes having highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105688.
Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105682.
Number of codes having highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105682.
Highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.
Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.
Number of inequivalent optimal Hermitian self-dual codes of length 2n over GF(4).
G.f.: (1+x^5+x^10)/((1-x)*(1-x^3)).
Number of indecomposable self-dual ternary codes of length 4n.
Highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n.
Highest minimal Hamming distance of any Type 3 ternary self-dual code of length 4n.
Highest minimal Hamming distance of any Type 4^E Euclidean linear self-dual code over GF(4) of length 2n.
Highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.
Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.
Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.
Number of inequivalent codes attaining highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.
Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.
Number of codes having highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.
Number of codes having highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.
Weight distribution of [24,12,6] odd Golay code.
Weight distribution of [13,6,6]_3 ternary code p_{13}.
Number of Type 4^E Euclidean self-dual codes over GF(4) of length 2n, excluding those of Type 4^E_{II}.
Number of Type 4^E_{II} Euclidean self-dual codes over GF(4) of length 4n.
Number of indecomposable Type II binary self-dual codes of length 8n.
Total number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n.
Number of indecomposable Type I but not Type II binary self-dual codes of length 2n.
Number of inequivalent (indecomposable or decomposable) Type I but not Type II binary self-dual codes of length 2n.
Number of indecomposable binary self-dual codes (singly- or doubly-even) of length 2n and minimal distance exactly 4.
Number of (indecomposable or decomposable) binary self-dual codes (singly- or doubly-even) of length 2n and minimal distance exactly 4.
Number of inequivalent codes attaining highest minimal Hamming distance of any Type (4_II)^H+ even Hermitian additive self-dual code over GF(4) of length 2n.
Number of (indecomposable or decomposable) binary self-dual codes (singly- or doubly-even) of length 2n and minimal distance exactly 6.
Number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n with the highest minimal distance.
Number of Type I (singly-even) self-dual binary codes of length 2n.
Number of decomposable binary self-dual codes of length 2n (up to permutation equivalence).