Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against pair errors in symbol-pair read channels. This special channel structure is motivated by the limitations of the reading process in high density data storage systems, where it is no longer possible to read individual symbols. In this work, we study bounds and constructions of codes in symbol-pair metric. By using some combinatorial structures, we give constructions of optimal q-ary symbol-pair codes with constant pair-weight <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1598_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$w_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and pair-distance <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1598_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$2w_p-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msub> <mi>w</mi> <mi>p</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for some length n, and some optimal q-ary codes with pair-weight <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1598_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$w_p=3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> for all pair-distance between 3 and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1598_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$2w_p-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msub> <mi>w</mi> <mi>p</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

Bounds and constructions of optimal symbol-pair codes with constant pair-weight

期刊论文
发表日期：

2025年2月22日

查看全文

Mengzhen Zhao,
Yanxun Chang

摘要

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against pair errors in symbol-pair read channels. This special channel structure is motivated by the limitations of the reading process in high density data storage systems, where it is no longer possible to read individual symbols. In this work, we study bounds and constructions of codes in symbol-pair metric. By using some combinatorial structures, we give constructions of optimal q-ary symbol-pair codes with constant pair-weight $w_p$ $w_{p}$ and pair-distance $2w_p-1$ $2 w_{p} - 1$ for some length n, and some optimal q-ary codes with pair-weight $w_p=3,4$ $w_{p} = 3, 4$ for all pair-distance between 3 and $2w_p-1$ $2 w_{p} - 1$ .