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Abstract: We study codes correcting adversarial insertions and deletions with list size $L$ fixed independently of the block length. We derive new achievable-rate bounds for binary codes and upper bounds over every fixed alphabet of size $q\ge2$, retaining explicit dependence on $L$. We establish a combinatorial reduction that trades $L$ units of insertion budget for one unit of deletion budget in the decoding guarantee, without changing the code or increasing the list size. Consequently, asymptotic bounds for mixed errors with insertion fraction $γ$ and deletion fraction $δ$ follow from insertion-only lower bounds at $γ+Lδ$ and deletion-only upper bounds at $δ+γ/L$. For binary unique decoding, we strictly improve Levenshtein's classical asymptotic rate lower bound for every deletion fraction $0<δ<1/2$ for which the classical rate expression is nonnegative. At $δ=0.1$, the lower bound increases from approximately $0.162009$ to $0.180431$, a relative increase of about $11.37\%$. Our framework also yields insertion and deletion lower bounds for every fixed list size. The existence proofs combine the Lovász local lemma with sampling from words having a specified number of runs, where a run is a maximal block of equal symbols. Generating functions provide refined bounds on the probability that $L+1$ sampled words share an allowed received word. We also derive a Levenshtein-type upper bound by run counting and, separately, a higher-order Elias bound using intersections and unions of the position sets used to embed $L+1$ codewords in a common supersequence. The latter recovers Yasunaga's asymptotic unique-decoding bound at $L=1$ and strictly improves the Haeupler--Shahrasbi--Sudan insertion bound for every fixed $L$ and $0<γ<q-1$. Numerical comparisons quantify the gains and remaining gaps.

Abstract: Let $q=3^m$, let $K=\mathbb F_{q^2}$, and let \[\mathcal T=\{x\in K^*:\operatorname{N}_{K/\mathbb F_q}(x)=1\}.\] For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely $\mathcal T$. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome $S$ and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of $S$ with respect to $\mathcal T$ and determine it exactly by the norm and the quadratic character of $\mathbb F_q$. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.

Abstract: This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \leq r < t$, a rate-$r/t$ convolutional code $\mathcal{C}_{\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$. Such a convolutional code $\mathcal{C}_{\text{convol}}$ is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width $n$. Each local code $\mathcal{C}_{\text{local}}$ of $\mathcal{C}_{\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} $\mathcal{C}_0$ in the code chain. The minimum distance $d_{\text{local}}$ of each local code of $\mathcal{C}_{\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\mathcal{C}_0$ in the code chain. The local structure of $\mathcal{C}_{\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\mathcal{C}_0$ using a sliding window decoding scheme.