Low Complexity Tail-Biting Trellises of Self-dual codes of Length 24, 32 and 40 over GF(2) and Z4 of Large Minimum Distance

We show in this article how the multi-stage encoding scheme proposed in [3] may be used to construct the [24] by using an extended [8, 4, 4] Hamming base code. An extension of the construction of [3] over Z4 yields self-dual codes over Z4 with parameters (for the Lee metric over Z4) [24, 12, 12] and [32, 16, 12] by using the [8, 4, 6] octacode. Moreover, there is a natural Tanner graph associated to the construction of [3], and it turns out that all our constructions have Tanner graphs that have a cyclic structure which gives tail-biting trellises of low complexity: 16-state tail-biting trellises for the [24, 12, 8], [32, 16, 8], [40, 20, 8] binary codes, and 256-state tail-biting trellises for the [24, 12, 12] and [32, 16, 12] codes over Z4.