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

Julia Lieb; Raquel Pinto

doi:10.3934/amc.2021031

Article Contents

2023, Volume 17, Issue 4: 935-959. Doi: 10.3934/amc.2021031

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

Julia Lieb^1, , and
Raquel Pinto^2,

1.
Institute of Mathematics, University of Zurich
2.
Department of Mathematics, University of Aveiro

^* Corresponding author: Julia Lieb
^* Corresponding author: Julia Lieb

Received: June 2020

Revised: March 2021

Early access: August 2021

Published: August 2023

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 94B10; Secondary: 94B35.

Citation:

Full Text(HTML)

$ \hat{v}_{ij} $	$ j=0 $	$ j=1 $	$ j=2 $	$ j=3 $	$ j=4 $
$ i=0 $	$ \ast $	$ v_{01,1} $	$ \ast $	$ \ast $	$ \ast $
	$ \ast $	$ v_{01,2} $	$ \ast $	$ \ast $	$ \ast $
	$ \ast $	$ v_{01,3} $	$ \ast $	$ v_{03,3} $	$ \ast $
$ i=1 $	$ \ast $	$ \ast $	$ v_{12,1} $	$ \ast $	$ v_{14,1} $
	$ \ast $	$ \ast $	$ v_{12,2} $	$ \ast $	$ v_{14,2} $
	$ \ast $	$ \ast $	$ v_{12,3} $	$ v_{13,3} $	$ v_{14,3} $
$ i=2 $	$ v_{20,1} $	$ \ast $	$ \ast $	$ \ast $	$ v_{24,1} $
	$ v_{20,2} $	$ \ast $	$ \ast $	$ \ast $	$ v_{24,2} $
	$ v_{20,3} $	$ v_{21,3} $	$ \ast $	$ v_{23,3} $	$ v_{24,3} $
$ i=3 $	$ \ast $	$ v_{31,1} $	$ \ast $	$ \ast $	$ v_{34,1} $
	$ \ast $	$ v_{31,2} $	$ \ast $	$ \ast $	$ v_{34,2} $
	$ \ast $	$ v_{31,3} $	$ \ast $	$ v_{33,3} $	$ v_{34,3} $
$ i=4 $	$ \ast $	$ v_{41,1} $	$ \ast $	$ \ast $	$ v_{44,1} $
	$ \ast $	$ v_{41,2} $	$ \ast $	$ \ast $	$ v_{44,2} $
	$ \ast $	$ v_{41,3} $	$ \ast $	$ v_{43,3} $	$ v_{44,3} $

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$ \hat{v}_{ij} $	$ j=0 $	$ j=1 $	$ j=2 $	$ j=3 $	$ j=4 $	$ j=5 $	$ j=6 $
$ i=0 $	$ v_{00,1} $	$ \ast $	$ v_{02,1} $	$ \ast $	$ v_{04,1} $	$ v_{05,1} $	$ v_{06,1} $
	$ v_{00,2} $	$ \ast $	$ v_{02,2} $	$ \ast $	$ \ast $	$ v_{05,2} $	$ v_{06,2} $
$ i=1 $	$ v_{10,1} $	$ \ast $	$ \ast $	$ \ast $	$ \ast $	$ v_{15,1} $	$ v_{16,1} $
	$ v_{10,2} $	$ \ast $	$ \ast $	$ \ast $	$ \ast $	$ v_{15,2} $	$ v_{16,2} $
$ i=2 $	$ \ast $	$ v_{21,1} $	$ v_{22,1} $	$ \ast $	$ \ast $	$ v_{25,1} $	$ v_{26,1} $
	$ \ast $	$ v_{21,2} $	$ v_{22,2} $	$ \ast $	$ \ast $	$ v_{25,2} $	$ v_{26,2} $
$ i=3 $	$ \ast $	$ \ast $	$ \ast $	$ \ast $	$ v_{34,1} $	$ v_{35,1} $	$ \ast $
	$ \ast $	$ \ast $	$ \ast $	$ \ast $	$ v_{34,2} $	$ v_{35,2} $	$ \ast $
$ i=4 $	$ v_{40,1} $	$ v_{41,1} $	$ v_{42,1} $	$ \ast $	$ \ast $	$ v_{45,1} $	$ v_{46,1} $
	$ v_{40,2} $	$ v_{41,2} $	$ v_{42,2} $	$ \ast $	$ \ast $	$ v_{45,2} $	$ v_{46,2} $
$ i=5 $	$ v_{50,1} $	$ v_{51,1} $	$ v_{52,1} $	$ \ast $	$ \ast $	$ v_{55,1} $	$ v_{56,1} $
	$ v_{50,2} $	$ v_{51,2} $	$ v_{52,2} $	$ \ast $	$ \ast $	$ v_{55,2} $	$ v_{56,2} $
$ i=6 $	$ v_{60,1} $	$ v_{61,1} $	$ v_{62,1} $	$ \ast $	$ \ast $	$ v_{65,1} $	$ v_{66,1} $
	$ v_{60,2} $	$ v_{61,2} $	$ v_{62,2} $	$ \ast $	$ \ast $	$ v_{65,2} $	$ v_{66,2} $

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$ \hat{v}_{ij} $	$ j=0 $	$ \cdots $	$ j=\deg_{z_1}(v(z_1,z_2)) $
$ i=0 $	$ \ast $	$ \cdots $	$ \ast $
$ \vdots $	$ \vdots $		$ \vdots $
$ i=(L_1+1)(n-k) $	$ \ast $	$ \cdots $	$ \ast $
$ i=(L_1+1)(n-k)+1 $	$ v_{(L_1+1)(n-k)+1,0} $	$ \cdots $	$ v_{(L_1+1)(n-k)+1,\deg_{z_1}(v(z_1,z_2))} $
$ \vdots $	$ \vdots $		$ \vdots $
$ i=(L_1+1)n $	$ v_{(L_1+1)n,0} $	$ \cdots $	$ v_{(L_1+1)n,\deg_{z_1}(v(z_1,z_2))} $

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