A decoding algorithm for 2D convolutional codes over the erasure channel

Abstract

<p style='text-indent:20px;'>Two-dimensional (2D) convolutional codes are a generalization of (1D) convolutional codes, which are suitable for transmission over an erasure channel. In this paper, we present a decoding algorithm for 2D convolutional codes over such a channel by reducing the decoding process to several decoding steps applied to 1D convolutional codes. Moreover, we provide constructions of 2D convolutional codes that are specially tailored to our decoding algorithm.</p>

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

“…where a j 0 = −H Ω ν v Ω ν is known and H W is as in (8). Moreover, consider the 1D convolutional code C h with a parity check as defined in (10) and with a column distance d ℓ , ℓ ≥ 0. If there exists ℓ 0 such that at most there are d c ℓ 0 − 1 erasures that occur in any (ℓ 0 + 1)n consecutive components of v W , then we can completely recover the vector v W .…”

“…Very recently, in [10], an alternative decoding algorithm for 2D convolutional codes was presented. The proposed method also follows the idea of decoding in two directions: horizontal and vertical.…”

“…if med = 0 then {Decides orientation depending on med} if length(R ori ) = ℓ + 1 then {All corrected} There are two main differences with respect to the algorithm described in this paper, the first of which is the decoding performance in a given line. While Algorithm 1 tries forward, backward, and complete-MDP (if possible) decoding for a given line, the method from [10] performs partial corrections that may leave gaps of erasures, whereby when it later it tries to fill those gaps it is always in the forward direction. This last-mentioned process of "filling the gaps" was performed by correcting one point at a time; however, this may make this algorithm less efficient in terms of time.…”

“…Example 8. Let us consider Example 3 in [10]. In Figure 5, the pattern of erasures ( * ) considered in that example is shown.…”

“…There exist only two decoding algorithms for 2D convolutional codes over the erasure channel [9,10]. In [9], the authors made use of the parity-check matrices of the code, while the authors of [10] employed the encoders of the code.…”