A new upper bound on the minimal distance of self-dual codes

Untitled

“…For example, consider the double circulant code D 4,6 of length 6 over Z 4 with 210 as the first row of R. This code has minimum Euclidean weight 6. Thus the isodual lattice A 4 (D 4,6 ) constructed from D 4,6 by Construction A has minimum norm 3 2 . The highest minimum norm among all known six-dimensional isodual lattices is 1 + 1 3 (=1.5773 .…”

“…Note that information on the highest minimum norm among isodual lattices in dimensions 17 to 22 is lacking in [5]. In this range of dimensions, the best known isodual lattices are modular lattices of level l. If such a lattice has minimum norm µ, then the corresponding isodual lattice has minimum norm µ/ √ l. We refer to the survey [11] for information on lattices with parameters: (n, l, µ) = (12, 3, 4), (14, 3, 4), (16, 2, 4), (18, 3,4) and (20,7,8).…”

“…The most interesting are the 22-dimensional isodual lattices with minimum norm 3 and kissing number 2464 constructed from the double circulant codes over Z 4 of length 22 and minimum Euclidean weight 12. In Section 5, we show that there are up to equivalence exactly six of these double circulant codes over Z 4 . We show that an extremal binary Type II code of length 24 can be constructed from each of these codes, pointing out a close connection between the 22-dimensional isodual lattices and the Leech lattice.…”

A Prize Problem in Coding Theory

“…This additional relation gives a further restriction on a possible weight enumerator of a binary self-dual code, often proving the nonexistence of a putative binary self-dual code [1].…”

“…Given a binary Type I code C, one can obtain the doubly-even subcode C 0 of C (consisting of all codewords whose weight ≡ 0 (mod 4)). The shadow S of C is defined by S := C ⊥ 0 \C [1]. The weight enumerator S(x, y) of the shadow of C is determined by the weight enumerator C(x, y) of C as S(x, y) =…”