Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

Abstract

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Gröbner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n 3 ) using Gaussian elimination to O(qn 2 ) with some mild conditions on n and q, where n is the code length and q is the finite-field size.

Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

“…To observe a precise complexity, we separate the decoding procedure into the error position determination and the error value determination. [22], [25] and O(qN 2 i ) = O(q 2m−2i+1 ) [15], respectively. According to [22], [25], we have z i ≤ N i /q = q m−i−1 < N i for all i ∈ {0, 1, .…”

“…We previously proposed a decoding algorithm [15,Algorithm 2] for a class of affine variety codes using the BMS algorithm and DFT. The following definitions are required to explain this decoding algorithm.…”

“…Error correction for C ⊥ (L, Ψ) [15] Input: (r P ) P∈Ψ ∈ F Ψ q , where (r P ) P∈Ψ = (c P ) P∈Ψ + (e P ) P∈Ψ , (c P ) P∈Ψ ∈ C ⊥ (L, Ψ) and (e P ) P∈Ψ ∈ F Ψ q Output: (ĉ P ) P∈Ψ…”

“…Finally, we compare the codeword error rate of our algorithm with that of the MDD and find them to be similar for some high-order PRM codes.The remainder of this paper is organized as follows. In Section 2, we present some preliminary notation and recall the results of a previous study [15]. In Section 3, we present an example of PRM code that shows a difficulty to construct a decoding algorithm.…”

“…Error word (e P ) P∈P 3 Ψ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 β 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Proof: For the Ψ i -component, the computational complexities of the error position determination and the error value determination are O(z i N 2 i ) = O(z i q 2m−2i ) [22], [25] and O(qN 2 i ) = O(q 2m−2i+1 ) [15], respectively. According to [22], [25], we have z i ≤ N i /q = q m−i−1 < N i for all i ∈ {0, 1, .…”