Abstract | ||
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Based on the extended Reed-Solomon (RS) code that has two information symbols over the field
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{q}$ </tex-math></inline-formula>
, we can construct a binary regular matrix
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${ {\boldsymbol{\textstyle H}}}(\gamma,\rho)$ </tex-math></inline-formula>
, where
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q:=2^{r}$ </tex-math></inline-formula>
for some positive integer
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>
,
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>
and
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>
are two positive integers such that
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma \le q$ </tex-math></inline-formula>
and
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho \le q$ </tex-math></inline-formula>
. The matrix
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${ {\boldsymbol{\textstyle H}}}(\gamma,\rho)$ </tex-math></inline-formula>
specifies a low-density parity-check (LDPC) code
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal C}(\gamma,\rho)$ </tex-math></inline-formula>
, called an RS-LDPC code. In this letter, we provide more results on the minimum distance and stopping distance of this class of codes (denoted by
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d({\mathcal C}(\gamma,\rho))$ </tex-math></inline-formula>
and
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({ {\boldsymbol{\textstyle H}}}(\gamma,\rho))$ </tex-math></inline-formula>
) for the case
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =6$ </tex-math></inline-formula>
. For
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho =q$ </tex-math></inline-formula>
, we derive an upper bound on
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({ {\boldsymbol{\textstyle H}}}(6,q))$ </tex-math></inline-formula>
and
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d(\mathcal {C}(6,q))$ </tex-math></inline-formula>
, which is conjectured to be tight for
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\ge 32$ </tex-math></inline-formula>
. For
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho < q$ </tex-math></inline-formula>
, we investigate the choices of
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>
such that
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d(\mathcal {C}(6,\rho))$ </tex-math></inline-formula>
(resp.,
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({ {\boldsymbol{\textstyle H}}}(6,\rho))$ </tex-math></inline-formula>
) can be improved compared with the original
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d(\mathcal {C}(6,q))$ </tex-math></inline-formula>
(resp.,
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s({ {\boldsymbol{\textstyle H}}}(6,q))$ </tex-math></inline-formula>
). |
Year | DOI | Venue |
---|---|---|
2020 | 10.1109/LCOMM.2019.2957802 | IEEE Communications Letters |
Keywords | DocType | Volume |
Upper bound,Linear codes,Signal to noise ratio,Generators,Indexes,Iterative decoding | Journal | 24 |
Issue | ISSN | Citations |
3 | 1089-7798 | 0 |
PageRank | References | Authors |
0.34 | 0 | 2 |
Name | Order | Citations | PageRank |
---|---|---|---|
Haiyang Liu | 1 | 28 | 10.84 |
Xiaopeng Jiao | 2 | 0 | 1.01 |