We study and propose schemes that map messages onto constant-weight codewords using variable-length prefixes. We provide polynomial-time computable formulas that estimate the average number of redundant bits incurred by our schemes. In addition to the exact formulas, we also perform an asymptotic analysis and demonstrate that our scheme uses $\frac12 \log n+O(1)$ redundant bits to encode messages into length-$n$ words with weight $(n/2)+{\sf q}$ for constant ${\sf q}$. We also propose schemes that map messages into balanced codebooks with error-correcting capabilities. For such schemes, we provide methods to enumerate the average number of redundant bits.

pdf/average-redundancy-of-variable-length-balancing-schemes-a-la-y4u6ga0g.pdf

Average Redundancy of Variable-Length Balancing Schemes à la Knuth

The study of constant-weight codes in $\ell _{1}$ -metric was motivated by the duplication-correcting problem for data storage in live DNA. It is interesting to determine the maximum size of a code given the length ${n}$ , weight ${w}$ , minimum distance ${d}$ and the alphabet size ${q}$ . In this paper, based on graph decompositions, we determine the maximum size of ternary codes with constant weight w and distance $2{w}-2$ for all sufficiently large length ${n}$ . Previously, this was known only for a very sparse family ${n}$ of density $4/{w}({w}-1)$ .

Optimal Ternary Codes With Weight w and Distance 2 w - 2 in ℓ 1 -Metric

The study of constant-weight codes in $\ell_1$-metric was motivated by the duplication-correcting problem for data storage in live DNA. It is interesting to determine the maximum size of a code given the length $n$, weight $w$, minimum distance $d$ and the alphabet size $q$. In this paper, based on graph decompositions, we determine the maximum size of ternary codes with constant weight $w$ and distance $2w-2$ for all sufficiently large length $n$. Previously, this was known only for a very sparse family $n$ of density $4/w(w-1)$.

Optimal Ternary Codes with Weight $w$ and Distance $2w-2$ in $\ell_1$-Metric

This paper studies the construction of constant <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-weight codes under <inline-formula> <tex-math notation="LaTeX">$L_{\infty }$ </tex-math></inline-formula>-metric, which could be used to design codes addressing 0-indels (i.e., the insertion/deletion of 0’s only). Based on the patterns of small codes, we determine the maximum size for codes with distance 2 and length 3 for any constant weight, and several constructions and algorithms for suboptimal codes with general length <inline-formula> <tex-math notation="LaTeX">$w+1$ </tex-math></inline-formula> are also provided. In general, for codes with distance D, length <inline-formula> <tex-math notation="LaTeX">$w+1$ </tex-math></inline-formula>, and weight <inline-formula> <tex-math notation="LaTeX">$tD+r$ </tex-math></inline-formula>, we derive the size of the largest code by using Hall’s theorem for integers <inline-formula> <tex-math notation="LaTeX">$r=0$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$t\in [{0,3}]$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$w\geq t-1$ </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">$r=\frac {D}{2}$ </tex-math></inline-formula> with D being even, codes with a cardinality that surpasses the known existing sizes by the additive term <inline-formula> <tex-math notation="LaTeX">$\left \lceil{ \frac {1}{3}\binom {t+1}{2}}\right \rceil \left \lfloor{ \frac {w+1}{3}}\right \rfloor $ </tex-math></inline-formula> are constructed based on the structure of group divisible designs in combinatorics.

Constant L
 1-weight codes under L
 ∞-metric

https://dr.ntu.edu.sg/bitstream/10356/165395/2/Dao%20Duc%20Tu_thesis.pdf

Applications of combinatorics in coding theory

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in $\ell _{1}$ -metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with $\ell _{1}$ -weight ${w}\leq 4$ for all possible distances. In general, we derive the size of the largest ternary code with constant weight $w$ and distance $2{w}-2$ for sufficiently large length n satisfying ${n}\equiv 1, {w},- {w}+2,-2{w}+3\pmod {{w}({w}-1)}$ .

Optimal Codes With Small Constant Weight in ℓ ₁-Metric

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in $\ell_1$-metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with $\ell_1$-weight $w\leq 4$ for all possible distances. In general, we derive the size of the largest ternary code with constant weight $w$ and distance $2w-2$ for sufficiently large length $n$ satisfying $n\equiv 1,w,-w+2,-2w+3\pmod{w(w-1)}$.

Optimal codes with small constant weight in $\ell_1$-metric

Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in $l_1$-metric. By combinatorial methods, we give constructions of optimal ternary codes with $l_1$-weight $w\leq 4$ for all possible distances. In general, we determine the maximum size of ternary codes with constant weight $w$ and distance $2w-2$ for sufficiently large length under certain conditions.

Optimal Constant-Weight Codes under l 1 -Metric.

Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions $xy^{-1}$ for all distinct two elements $x,y\in D$, represent each non-identity element in $G$ exactly $\lambda$ times. A $(v,k,\lambda)$-signed difference set is a generalization of a $(v,k,\lambda)$-difference set $D$, which satisfies all properties of $D$, but has a sign for each element in $D$. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes.

pdf/new-constructions-of-signed-difference-sets-1dheijsa.pdf

New constructions of signed difference sets

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