q-Analogs of Packing Designs

Braun, Michael; Reichelt, Jan

doi:10.48550/arxiv.1212.4614

Cited by 4 publications

(16 citation statements)

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“…We close this section with two examples. The first example recalls the construction of a (7, 301, 4; 3) 2 code in [36,29], which was used as the intermediate step in an alternative construction of a currently best known (7, 329, 4; 3) 2 code in [36,29], following the original discovery of such a code in [11].…”

Section: The Refined Approachmentioning

confidence: 99%

“…36 One can show that L n has no zero divisors and admits one-sided analogues of the Euclidean Algorithm for symbolic division. The center of L n consists of all polynomials of the form c 0 37 The computation of subspace polynomials is facilitated by the following symbolic factorization into linear factors: (11) s…”

Section: Dickson Invariants Subspace Polynomials and All Thatmentioning

confidence: 99%

“…In the smallest open case v = 7 we know 329 ≤ A 2 (7, 4; 3) ≤ 381. The lower bound stems from the computer-aided group-invariant construction in [11] (cf. [36,29] for alternative constructions), and the upper bound is the size of a putative 2-analogue of the projective plane of order 2, whose existence is still undecided (despite the known solution in the much larger case v = 13).…”

Section: Introductionmentioning

confidence: 99%

“…The subspace code optimization algorithm should consider all such combinations and select the best one. With these two guidelines at hand, we created a simple prototype of the subspace code optimization algorithm, which generated planes W randomly, evaluated the associated invariant σ W on all planes intersecting W in a line, 10 and computed an optimal solution of the resulting optimization problem-maximize the difference between the image size of σ W and the (suitably normalized 11 ) number of planes removed from L W , subsequently referred to as the local net gain-in a brute-force manner by exhaustive search through all combinations of minimal subsets of G W . With this algorithm and the additional observation that the problem setting is invariant under a fairly large group of collineations of PG(F 2 n ) acting on the set of planes W , we were able to solve the cases v = 8, 9, 10 completely and do a partial search for length v = 11.…”

Section: Introductionmentioning

confidence: 99%

“…10 The domain of σ W consists precisely of those planes. 11 The total number of planes removed from L W is divided by the number of points on the special flat S defined in Section 2, in order to match the present "local" point-of-view.…”

Section: Introductionmentioning

confidence: 99%

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The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

Ai¹,

Honold²,

Liu³

2016

Preprint

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As shown in [31], one of the five isomorphism types of optimal binary subspace codes of size 77 for packet length v = 6, constant dimension k = 3 and minimum subspace distance d = 4 can be constructed by first expurgating and then augmenting the corresponding lifted Gabidulin code in a fairly simple way. The method was refined in [36,29] to yield an essentially computer-free construction of a currently best-known plane subspace code of size 329 for (v, k, d) = (7, 3, 4). In this paper we generalize the expurgationaugmentation approach to arbitrary packet length v, providing both a detailed theoretical analysis of our method and computational results for small parameters. As it turns out, our method is capable of producing codes larger than those obtained by the echelon-Ferrers construction and its variants. We are able to prove this observation rigorously for packet lengths v ≡ 3 (mod 4).

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Section: The Refined Approachmentioning

confidence: 99%

Section: Dickson Invariants Subspace Polynomials and All Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

Ai¹,

Honold²,

Liu³

2016

Preprint

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Poster: A new approach to the Main Problem of Subspace Coding

Liu

Honold

2014

9th International Conference on Communications and Networking in China

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Subspace codes form the appropriate mathematical setting for investigating the Koetter-Kschischang model of fault-tolerant network coding. The Main Problem of Subspace Coding asks for the determination of a subspace code of maximum size (proportional to the transmission rate) if the remaining parameters are kept fixed. We describe a new approach to finding good subspace codes, which surpasses the known size limit of lifted MRD codes and is capable of yielding an alternative construction of the currently best known binary subspace code of packet length 7, constant dimension 3 and minimum subspace distance 4.

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New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

Chen

Weng

et al. 2020

IEEE Trans. Inform. Theory

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A basic problem for the constant dimension subspace coding is to determine the maximal possible size A q (n, d, k) of a set of k-dimensional subspaces in F n q such that the subspace distance satisfies d(U, V ) = 2k − 2 dim(U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximal rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem of the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A q (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

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q-Analogs of Packing Designs

Cited by 4 publications

References 0 publications

The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

Poster: A new approach to the Main Problem of Subspace Coding

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

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