Bounds on the state complexity of codes from the Hermitian function field and its subfields

Abstract

An upper bound on the minimal state complexity of codes from the Hermitian function field and some of its subfields is derived. Coordinate orderings under which the state complexity of the codes is not above the bound are specified. For the self-dual Hermitian code it is proved that the bound coincides with the minimal state complexity of the code. Finally, it is shown that Hermitian codes over fields of characteristic 2 admit a recursive twisted squaring construction.Abstract-A class of linear codes with good parameters is constructed in this correspondence. It turns out that linear codes of this class are subcodes of the subfield subcodes of Goppa's geometry codes. In particular, we find 61 improvements on Brouwer's table [1] based on our codes.

Determining When the Absolute State Complexity of a Hermitian Code Achieves Its DLP Bound

“…Thus P 2 gain (13) = {5, 8, 6, 11, 9, 7, 14} = [5, 9]∪{11, 14} and P gain (13) = [1, 9]∪{11, 14}, and similarly for P fall (13). We have P gain (13) < P fall (13) and so s(C 13 ) = 11.…”

“…Comparing these values of s(C m ) with the values of Table 2 we have s( Table 2 is not in boldface. We remark that the main result of [13] is Example 5.11 with q ≥ 4. Corollary 5.10 and Proposition 3.12 imply that ∇(C m ) is attained for just over half the m ∈ I(n, g).…”

“…For a stronger version of [13,Proposition 1] (an application of Clifford's theorem), see [5,Proposition 3.4]. Also, Example 5.11 below generalizes the main result of [13] to arbitrary self-dual Hermitian codes. An initial account of some of these results appeared in [8].…”

“…The state complexity of Hermitian codes has also been studied in [13]. For a stronger version of [13,Proposition 1] (an application of Clifford's theorem), see [5,Proposition 3.4].…”

Bounding the Trellis State Complexity of Algebraic Geometric Codes

“…Here we just mention the Wolf bound, as it was first noticed by him in [22], namely s(C) ≤ w(C) := min{k, n − k} , where k is the dimension of C. The study of the state complexity of some classical codes, such as BCH, RS, and RM codes, has been carried out by several authors; see [2], [3], [4], [11], [21]. The case of algebraic geometric codes (or simply, AG codes) was treated by Shany and Be'ery [18], Blackmore and Norton [5], [6], and by Munuera and Torres [15]. If C = C(X , D, G) is an AG code, from these works it follows that s(C) = w(C), provided that either deg(G) < ⌊deg(D)/2⌋, or deg(G) > ⌈deg(D)/2⌉ + 2g − 2, with g being the genus of the underlying curve X .…”