4,440 research outputs found

    Subspace Designs Based on Algebraic Function Fields

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    Subspace designs are a (large) collection of high-dimensional subspaces {H_i} of F_q^m such that for any low-dimensional subspace W, only a small number of subspaces from the collection have non-trivial intersection with W; more precisely, the sum of dimensions of W cap H_i is at most some parameter L. The notion was put forth by Guruswami and Xing (STOC\u2713) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS\u2713, Combinatorica\u2716) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically q >= m). Forbes and Guruswami (RANDOM\u2715) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound LL on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over F^n for any field F, with logarithmic degree and expansion guarantee for subspaces of dimension Omega(n/(log(log(n))))

    Problems on q-Analogs in Coding Theory

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    The interest in qq-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

    Higgledy-piggledy subspaces and uniform subspace designs

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    In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective kk-subspaces in PG(d,F)\mathsf{PG}(d,\mathbb{F}) are in `higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension kk in a generator set of points. We prove that the set H\mathcal{H} of higgledy-piggledy kk-subspaces has to contain more than min⁑∣F∣,βˆ‘i=0k⌊dβˆ’k+ii+1βŒ‹\min{|\mathbb{F}|,\sum_{i=0}^k\lfloor\frac{d-k+i}{i+1}\rfloor} elements. We also prove that H\mathcal{H} has to contain more than (k+1)β‹…(dβˆ’k)(k+1)\cdot(d-k) elements if the field F\mathbb{F} is algebraically closed. An rr-uniform weak (s,A)(s,A) subspace design is a set of linear subspaces H1,..,HN≀FmH_1,..,H_N\le\mathbb{F}^m each of rank rr such that each linear subspace W≀FmW\le\mathbb{F}^m of rank ss meets at most AA among them. This subspace design is an rr-uniform strong (s,A)(s,A) subspace design if βˆ‘i=1Nrank(Hi∩W)≀A\sum_{i=1}^N\mathrm{rank}(H_i\cap W)\le A for βˆ€W≀Fm\forall W\le\mathbb{F}^m of rank ss. We prove that if m=r+sm=r+s then the dual ({H1βŠ₯,...,HNβŠ₯}\{H_1^\bot,...,H_N^\bot\}) of an rr-uniform weak (strong) subspace design of parameter (s,A)(s,A) is an ss-uniform weak (strong) subspace design of parameter (r,A)(r,A). We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that Aβ‰₯min⁑∣F∣,βˆ‘i=0rβˆ’1⌊s+ii+1βŒ‹A\ge\min{|\mathbb{F}|,\sum_{i=0}^{r-1}\lfloor\frac{s+i}{i+1}\rfloor} for rr-uniform weak or strong (s,A)(s,A) subspace designs in Fr+s\mathbb{F}^{r+s}. We show that the rr-uniform strong (s,rβ‹…s+(r2))(s,r\cdot s+\binom{r}{2}) subspace design constructed by Guruswami and Kopprty (based on multiplicity codes) has parameter A=rβ‹…sA=r\cdot s if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound (k+1)β‹…(dβˆ’k)+1(k+1)\cdot(d-k)+1 over algebraically closed field is tight.Comment: 27 pages. Submitted to Designs Codes and Cryptograph

    On some representations of quadratic APN functions and dimensional dual hyperovals

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