arXiv subjects

Tim Alderson

Publications and source records attributed to Tim Alderson.

5 recordsLinked to original sources

arXiv2026

On the Maximality of Additive Codes

An additive $(n,k,d)_{q^m/q}$-code is a $\mathrm{GF}(q)$-linear subspace of $\mathrm{GF}(q^m)^n$ of $\mathrm{GF}(q)$-dimension $km$ with minimum Hamming distance $d$. We first extend the Alderson--Bruen--Silverman (ABS) model of linear codes to the additive setting: a code of length $n$ with $q^{km}$ words over an alphabet of size $q^m$ admits an ABS model if and only if it is equivalent to a nondegenerate additive code. We then ask whether an additive code that admits an extension must admit an \emph{additive} extension. For linear codes ($m=1$) this is a theorem of Alderson and G\'acs. We characterize the additive codes admitting no additive extension as those whose associated projective system of flats is complete, and we prove that the answer to the question above is again affirmative for $(n,2,d)_{9/3}$-, $(n,2,d)_{4/2}$-, and $(n,3,d)_{4/2}$-codes. In contrast with the linear case, we show that the answer is negative in general. Scattered linear sets yield, for each square $q$, extendable additive $(n,2,d)_{q^2/q}$-codes admitting no additive extension. Further, a different method yields an extendable additive $(30,2,24)_{8/2}$-code with no additive extension. Consequently, for properly additive codes, completeness of the associated projective system does not imply maximality of the code. We conjecture that extendable $(n,2,d)_{p^2/p}$-codes, $p$ prime, always admit additive extensions.

cs.IT↗

arXiv2026

Length-Maximal Codes with Given Singleton Defect: Structure and Bounds

Let $\mathcal C$ be an $(n,q^k,d)_q$ code of integer dimension $k\ge2$ and Singleton defect $s=n-k+1-d$. Repeated shortening followed by the Plotkin bound gives \[ n\le (s+1)(q+1)+k-2. \] We study the equality case, calling a code attaining this bound \emph{length-maximal}. Equality provides rigid structure: the code is symbol-uniform under every successive shortening, is an orthogonal array of strength at least $k-1$, and has pairwise distances in $\{d\}\cup\{n-k+3,\ldots,n\}$. For $k\ge3$ it also satisfies $s\le q-1$ and $(s+2)\mid q(q+1)$. In dimension two, length-maximal codes are equivalent to resolvable $2$-$(q^2,q,s+1)$ multidesigns. The (inner) distance distribution, and hence the Hamming weight enumerator after any codeword is normalised to zero, is determined by the parameters. In dimension four this gives three cases: apart from the MDS case, only $s=q-2$ and $s=q-1$ remain possible; in the latter case the alphabet size is constrained by $(q+2)\mid36$, leaving six values of $q$. For $q\ge4$, every length-maximal code of dimension at least five is MDS, and requires $q$ to be a multiple of $36$. The non-MDS cases are the dual and extended ternary Golay codes, of dimensions five and six. Consequently, for $s\ge1$ one has $k\le4$ unless $q=3$, where $k\le6$. By shortening the codes before applying Plotkin, we provide sharper large-defect bounds and several ranges in which nonlinear codes satisfy the Griesmer bound.

math.CO↗

arXiv2026

Sets of subspaces with restricted hyperplane intersection numbers

Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and characterise the structure of $\mathcal{X}$ in the case of equality. We call spanning sets attaining this bound \emph{length-maximal}. For $k=3$, these sets are higher-dimensional analogues of maximal arcs; when $h=1$ they are precisely the maximal arcs of $\mathrm{PG}(2,q)$, which for $t 2$, we show that any length-maximal set must satisfy $t = q^h+1$ and that every hyperplane is either a $t$-secant or a $1$-secant. Such sets exist for all $q$ and $h$, arising as field reductions of ovoids of $\mathrm{PG}(3,q^h)$. For $k \geq 5$ and $q^h>3$, no length-maximal set exists. The case $q^h=3$, that is $(q,h)=(3,1)$, is exceptional: there, examples arising from the ternary Golay code exist for $k=5$ and $k=6$, while none exist for $k\ge7$. In the language of additive codes, these results assert that additive codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q^h>3$, apart from these two sporadic $\mathbb{F}_3$ examples. We also determine the strongly regular graphs associated with length-maximal sets.

math.CO↗

arXiv2022

MWS and FWS Codes for Coordinate-Wise Weight Functions

A combinatorial problem concerning the maximum size of the (hamming) weight set of an $[n,k]_q$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $[n,k]_q $ codes with the same weight set as $ \mathbb{F}_q^n $ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed $ k,q $ the values of $ n $ for which an $ [n,k]_q $-FWS code exists are completely determined, but the determination of the minimum length $ M(H,k,q) $ of an $ [n,k]_q $-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on $ n $ for which an FWS code exists, and bounds on $ n $ for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $ M(\mathcal{L},k,q) $ (the minimum length of Lee MWS codes), and pose the determination of $ M(\mathcal{L},k,q) $ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.

cs.IT↗

arXiv2018

n-Dimensional Optical Orthogonal Codes, Bounds and Optimal Constructions

We generalized to higher dimensions the notions of optical orthogonal codes. We establish uper bounds on the capacity of general $ n $-dimensional OOCs, and on specific types of ideal codes (codes with zero off-peak autocorrelation). The bounds are based on the Johnson bound, and subsume many of the bounds that are typically applied to codes of dimension three or less. We also present two new constructions of ideal codes; one furnishes an infinite family of optimal codes for each dimension $ n\ge 2 $, and another which provides an asymptotically optimal family for each dimension $ n\ge 2 $. The constructions presented are based on certain point-sets in finite projective spaces of dimension $k$ over $GF(q)$ denoted $PG(k,q)$.

math.CO↗