Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

New bounds for the minimum length of quaternary linear codes of dimension five

AbstractLet n4(k, d) be the smallest integer n, such that a quaternary linear [n, k, d]-code exists. The bounds n4(5, 21) ⩽ 32, n4(5, 30) = 43, n4(5, 32) = 46, n4(5, 36) = 51, n4(5,40) ⩽ 57, n4(5, 48) ⩽ 67, n4(5, 64) = 88, n4(5, 68) ⩽ 94, n4(5, 70) ⩽ 97, n4(5, 92) ⩽ 126, n4(5, 98) ⩽ 135, n4(5, 122) = 165, n4(5, 132) ⩽ 179, n4(5, 136) ⩽ 184, n4(5, 140) = 189, n4(5, 156) ⩽ 211, n4(5,162) = 219, n4(5, 164) ⩽ 222, n4(5, 166) ⩽ 225, n4(5, 173) ⩽ 234, n4(5, 194) = 261, n4(5, 204) = 273, n4(5, 208) = 279, n4(5, 212) = 284, n4(5, 214) = 287, n4(5, 216) = 290 and n4(5, 220) = 295 are proved. A [q4 + q2 + 1, 5, q4 − q3 + q2 − q]-code over GF(q) exists for every q

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Quantum Error Correction via Codes over GF(4)

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information Theory. Replaced Sept. 24, 1996, to correct a number of minor errors. Replaced Sept. 10, 1997. The second section has been completely rewritten, and should hopefully be much clearer. We have also added a new section discussing the developments of the past year. Finally, we again corrected a number of minor error

Ganzzahlige quadratische Formen und Gitter

[no abstract available