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

Abstract

Abstract. Let g be the genus of the Hermitian function field H/F q 2 and let C L (D, mQ∞) be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of C L (D, mQ∞). Here we determine when this lower bound is tight and when it is not.For+ 2g, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of C L (D, mQ∞), which improves the DLP lower bound.Next we give a "good" coordinate order for C L (D, mQ∞). With this good order, the state complexity of C L (D, mQ∞) achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of C L (D, mQ∞) (for those values of m for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases.A straightforward application of these results is that if , and, in particular, so does its absolute state complexity.

Bounding the Trellis State Complexity of Algebraic Geometric Codes

“…The dual of C is C ⊥ = C(X , D, D + W − G), where W is a canonical divisor obtained as the divisor of a differential form having simple poles and residue 1 at every point in supp(D). Thus we deduce that C is formally self-orthogonal if there is an effective divisor E such that supp(E) ∩ supp(D) = ∅ and D + W − 2G ∼ E. Now, let C = C(X , D, G) be a Hermitian code, that is, a code constructed from a Hermitian curve of affine equation y q +y = x q+1 over F q 2 , by taking Q := (0 : 1 : 0), D equals to the sum of the q 3 affine points, and G = mQ (see [5], [23]). Since D + W − 2G ∼ (n + 2g − 2 − 2m)Q, then C is self-orthogonal whenever its dimension is at most n/2.…”

“…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 .…”

“…The particular case of Hermitian codes have been treated in [5], [6] and [18]. The purpose of this paper is to investigate further lower bounds on s(C) for C = C(X , D, G) an AG code.…”

“…In [5] this bound was computed by the same authors for the case of Hermitian codes and used to give a very good bound on s(C) for these codes. By using specific properties of R (cf.…”

“…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]. The main result in [15] is a Goppa-like bound on s(C), s(C) ≥ w(C) − (g − α) with α being the abundance of the code.…”