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Bounds and Constructions of Codes for Ordered Composite DNA Sequences

Zuo Ye, Yuling Li, Zhaojun Lan, Gennian Ge

TL;DR

This paper establishes equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs) and derives novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods.

Abstract

This paper extends the foundational work of Dollma \emph{et al}. on codes for ordered composite DNA sequences. We consider the general setting with an alphabet of size $q$ and a resolution parameter $k$, moving beyond the binary ($q=2$) case primarily studied previously. We investigate error-correcting codes for substitution errors and deletion errors under several channel models, including $(e_1,\ldots,e_k)$-composite error/deletion, $e$-composite error/deletion, and the newly introduced $t$-$(e_1,\ldots,e_t)$-composite error/deletion model. We first establish equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs). This significantly reduces the number of distinct error-parameter sets that require separate analysis. We then derive novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods. These bounds together cover all values of parameters $q$, $k$, $(e_1,\ldots,e_k)$ and $e$. In contrast, previous bounds were only established for $q=2$ and limited choices of $k$, $(e_1,\ldots,e_k)$ and $e$. For CDCCs, we generalize a known non-asymptotic upper bound for $(1,0,\ldots,0)$-CDCCs and then provide a cleaner asymptotic bound. On the constructive side, for any $q\ge2$, we propose $(1,0,\ldots,0)$-CDCCs, $1$-CDCCs and $t$-$(1,\ldots,1)$-CDCCs with near-optimal redundancies. These codes have efficient and systematic encoders. For substitution errors, we design the first explicit encoding and decoding algorithms for the binary $(1,0,\ldots,0)$-CECC constructed by Dollma \emph{et al}, and extend the approach to general $q$. Furthermore, we give an improved construction of binary $1$-CECCs, a construction of nonbinary $1$-CECCs, and a construction of $t$-$(1,\ldots,1)$-CECCs. These constructions are also systematic.

Bounds and Constructions of Codes for Ordered Composite DNA Sequences

TL;DR

This paper establishes equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs) and derives novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods.

Abstract

This paper extends the foundational work of Dollma \emph{et al}. on codes for ordered composite DNA sequences. We consider the general setting with an alphabet of size and a resolution parameter , moving beyond the binary () case primarily studied previously. We investigate error-correcting codes for substitution errors and deletion errors under several channel models, including -composite error/deletion, -composite error/deletion, and the newly introduced --composite error/deletion model. We first establish equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs). This significantly reduces the number of distinct error-parameter sets that require separate analysis. We then derive novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods. These bounds together cover all values of parameters , , and . In contrast, previous bounds were only established for and limited choices of , and . For CDCCs, we generalize a known non-asymptotic upper bound for -CDCCs and then provide a cleaner asymptotic bound. On the constructive side, for any , we propose -CDCCs, -CDCCs and --CDCCs with near-optimal redundancies. These codes have efficient and systematic encoders. For substitution errors, we design the first explicit encoding and decoding algorithms for the binary -CECC constructed by Dollma \emph{et al}, and extend the approach to general . Furthermore, we give an improved construction of binary -CECCs, a construction of nonbinary -CECCs, and a construction of --CECCs. These constructions are also systematic.
Paper Structure (24 sections, 41 theorems, 84 equations)

This paper contains 24 sections, 41 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Upper Bounds
  4. Constructions
  5. A New Error Type
  6. Preliminary
  7. Ordered Composite Letters and Sequences
  8. The Ordered Composite DNA Channel and Error Models
  9. Code Equivalence
  10. Upper Bounds on Code Sizes
  11. Bounds on Codes Correcting Substitutions
  12. The case $m>q$
  13. Comparison with previous results
  14. Bounds on Codes Correcting Deletions
  15. Constructions of Codes Correcting Deletions
  16. ...and 9 more sections

Key Result

Proposition 3.1

For any $\left( e_1,e_2,\ldots,e_{k} \right)\in\mathbb{N}^k$, the families $\mathscr{C}_{q,k}^S\left( n;e_1,e_2,\ldots,e_{k} \right)$ and $\mathscr{C}_{q,k}^S\left( n;e_{k},e_{k-1},\ldots,e_1 \right)$ are equivalent.

Theorems & Definitions (55)

  • Example 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Corollary 3.1
  • Remark 4.1
  • ...and 45 more