On Trellis Structures for Reed–Muller Codes

Abstract

We study trellises of Reed}Muller codes from "rst principles. Our approach to local trellis behaviour seems to be new and yields amongst other things another proof of a result of Berger and Be'ery on the state complexity of Reed}Muller codes. We give a general form of a minimal-span generator matrix for the family of Reed}Muller codes with their standard bit-order. We apply this to determining the number of parallel subtrellises in any uniform sectionalisation of a Reed}Muller code and to designing trellises for Reed}Muller codes with more parallel subtrellises than the minimal trellis, but with the same state complexity. Academic Press

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

“…We note that there are k points of gain and k points of fall. Points of gain and fall describe the local behavior of a minimal trellis [6], and being able to give a succinct characterization of them for particular families of codes has been useful in calculating formulae for their state complexity; see, e.g., [3,6]. The same proves to be the case here.…”

“…Points of gain and fall were introduced in [3,6]. For this paragraph, C is a length n linear code with dimension k. We note that dim(C i,− ) (as defined in section 2) increases in unit steps from 0 to k and dim(C i,+ ) decreases in unit steps from k to 0 as i increases from 0 to n.…”

“…"Points of gain and fall" were introduced in [3,4,6,7] to help determine the state complexity of certain generalizations of Reed-Muller codes. For these codes, the points of gain and fall had particularly nice characterizations.…”

On the treewidth of MDS and Reed-Muller codes

“…To put this another way, the branch complexity of the minimal trellis representation of RM(r, m) in the standard bit order attains the lower bound on, and thus equals, the trelliswidth of the code. Techniques from [2] allow us to compute the branch complexity of this trellis representation; for details, see [11]. We then have the following result.…”

Bounding the Trellis State Complexity of Algebraic Geometric Codes

“…5.2,5.3]. 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].…”