Dual code

code E6 over an Abelian group A4 with four elements is presented. E6 is cyclic, unlike the [6,3,4] hexacode HG over GF(4). However, E 6 and Hfi are isomorphic when the latter is viewed as a group code. Differences and similarities between E6 and H 6 are discussed. A dual code of E6 is presented. Some binary codes, among them the [24.12.8]Golay, are derived with the aid of E6 A related cyclic [1 2 31 code E; is applied to construct the Nordstrom-Robinson code. €6 is the smallest member of a class of [2k, k , 41 cyclic and reversible codes over Aq Another class of cyclic and reversible codes of length 21 + 1 E 2 2 and minimum distance 3 over A4 is also presented.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.