arXiv subjects

Tim L. Alderson

Publications and source records attributed to Tim L. Alderson.

7 recordsLinked to original sources

arXiv2025

When Arcs Extend Uniquely: A Higher-Dimensional Generalization of Barlotti's Result

In this short communication, we generalize a classical result of Barlotti concerning the unique extendability of arcs in the projective plane to higher-dimensional projective spaces. Specifically, we show that for integers $ k \ge 3 $, $ s \ge 0 $, and prime power $ q $, any $(n, k + s - 1)$-arc in PG$(k - 1, q)$ of size $ n = (s+1)(q+1) + k - 3 $ admits a unique extension to a maximal arc, provided $ s + 2 \mid q $ and $ s < q - 2 $. This result extends the classical characterizations of maximal arcs in PG$(2,q)$ and connects naturally to the theory of A$^s$MDS codes. Our findings establish conditions under which linear codes of given dimension and Singleton defect can be uniquely extended to maximal-length projective codes.

math.CO↗

arXiv2025

Projective systems and bounds on the length of codes of non-zero defect

We derive bounds on the lengths of linear codes with fixed Singleton defect $s$, working within the framework of projective systems as advocated by Tsfasman and Vl\v{a}du\c{t}. This geometric perspective allows us to unify and extend a range of existing results. We introduce the parameter $m^s(k,q)$, denoting the maximum length of a non-degenerate $[n,k,d]_q$ A$^s$MDS code, and more generally $m^s_t(k,q)$, where the dual code is additionally required to be A$^t$MDS. We also study $\kappa(s,q)$, the maximum dimension $k$ for which a length-maximal A$^s$MDS code exists. Among our main results, we provide sufficient conditions on $n$ and $k$ under which the dual of an A$^s$MDS code is necessarily A$^s$MDS, addressing a gap in the existing literature. We show that codes of sufficient length must be projective, meet the Griesmer bound, and be dual to an AMDS code. Our bounds subsume or improve several results in the literature. Two conjectures on the non-existence of length-maximal codes of dimension $k\ge 5$ are proposed, supported by computational evidence.

math.CO↗

arXiv2024

Bounds on MLDR Codes Over ${\mathbb Z}_{p^t}$

Upper bounds on the minimum Lee distance of codes that are linear over ${\mathbb Z}_q$, $q=p^t$, $p$ prime are discussed. The bounds are Singleton like, depending on the length, rank, and alphabet size of the code. Codes meeting such bounds are referred to as Maximum Lee Distance with respect to Rank (MLDR) Codes. We present some new bounds on MLDR codes, using combinatorial arguments. In the context of MLDR codes, our work provides improvements over existing bounds in the literature

math.CO↗

arXiv2019

On the Weights of General MDS Codes

The weight spectra of MDS codes of length $ n $ and dimension $ k $ over the arbitrary alphabets are studied. For all $ q $-ary MDS codes of dimension $ k $ containing the zero codeword, it is shown that all $ k $ weights from $ n $ to $ n-k+1 $ are realized. The remaining case $ n=q+k-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.

math.CO↗

arXiv2019

A note on full weight spectrum codes

A linear $ [n,k]_q $ code $ C $ is said to be a full weight spectrum (FWS) code if there exist codewords of each nonzero weight less than or equal to $ n $. In this brief communication we determine necessary and sufficient conditions for the existence of linear $ [n,k]_q $ full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in $ PG(k-1,q) $.

cs.IT↗

arXiv2018

Maximum Weight Spectrum Codes

In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field $\F_q$ was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by $\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $ q=2 $, and in the case $ k=2 $. They conjectured that the bound is sharp for every prime power $ q $ and every positive integer $ k $. In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.

cs.IT↗

arXiv2017

3-Dimensional Optical Orthogonal Codes with Ideal Autocorrelation-Bounds and Optimal Constructions

Several new constructions of 3-dimensional optical orthogonal codes are presented here. In each case the codes have ideal autocorrelation $\mathbf{ λ_a=0} $, and in all but one case a cross correlation of $ \mathbf{λ_c=1} $. All codes produced are optimal with respect to the applicable Johnson bound either presented or developed here. Thus, on one hand the codes are as large as possible, and on the other, the bound(s) are shown to be tight. All codes are constructed by using a particular automorphism (a Singer cycle) of $ \mathbf{ PG(k,q)} $, the finite projective geometry of dimension $ k $ over the field of order $ \mathbf{q} $, or by using an affine analogue in $ AG(k,q) $.

cs.IT↗