264 research outputs found
Algebraic Geometry in Coding Theory and Cryptography
No full textThis textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algeThis textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like al
Subspace Designs Based on Algebraic Function Fields
No full textSubspace 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'13) 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'13, Combinatorica'16) 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'15) 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 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))))
Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph
No full textLocally recoverable codes are a class of block codes with an additional property called locality. A locally recoverable code with locality r can recover a symbol by reading at most r other symbols. Recently, it was discovered by several authors that a q-ary optimal locally recoverable code, i.e., a locally recoverable code achieving the Singleton-type bound, can have length much bigger than q+1. In this paper, we present both the upper bound and the lower bound on the length of optimal locally recoverable codes. Our lower bound improves the best known result in [Yuan Luo et al., 2018] for all distance d >= 7. This result is built on the observation of the parity-check matrix equipped with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure produces an optimal locally recoverable code if it satisfies a certain expansion property for subsets of F_q. To our surprise, this expansion property is then shown to be equivalent to a well-studied problem in extremal graph theory. Our upper bound is derived by an refined analysis of the arguments of Theorem 3.3 in [Venkatesan Guruswami et al., 2018]
List Decoding of Rank-Metric Codes with Row-To-Column Ratio Bigger Than 1/2
No full textDespite numerous results about the list decoding of Hamming-metric codes, development of list decoding on rank-metric codes is not as rapid as its counterpart. The bound of list decoding obeys the Gilbert-Varshamov bound in both the metrics. In the case of the Hamming-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and alphabet size, while in the case of the rank-metric, the Gilbert-Varshamov bound is a trade-off among rate, decoding radius and column-to-row ratio (i.e., the ratio between the numbers of columns and rows). Hence, alphabet size and column-to-row ratio play a similar role for list decodability in each metric. In the case of the Hamming-metric, it is more challenging to list decode codes over smaller alphabets. In contrast, in the case of the rank-metric, it is more difficult to list decode codes with large column-to-row ratio. In particular, it is extremely difficult to list decode square matrix rank-metric codes (i.e., the column-to-row ratio is equal to 1).
The main purpose of this paper is to explicitly construct a class of rank-metric codes of rate R with the column-to-row ratio up to 2/3 and efficiently list decode these codes with decoding radius beyond the decoding radius (1-R)/2 (note that (1-R)/2 is at least half of relative minimum distance δ). In literature, the largest column-to-row ratio of rank-metric codes that can be efficiently list decoded beyond half of minimum distance is 1/2. Thus, it is greatly desired to efficiently design list decoding algorithms for rank-metric codes with the column-to-row ratio bigger than 1/2 or even close to 1. Our key idea is to compress an element of the field F_qⁿ into a smaller F_q-subspace via a linearized polynomial. Thus, the column-to-row ratio gets increased at the price of reducing the code rate. Our result shows that the compression technique is powerful and it has not been employed in the topic of list decoding of both the Hamming and rank metrics. Apart from the above algebraic technique, we follow some standard techniques to prune down the list. The algebraic idea enables us to pin down the message into a structured subspace of dimension linear in the number n of columns. This "periodic" structure allows us to pre-encode the message to prune down the list
Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs
No full textFor a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal.
We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:
- Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n).
- Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree
Constructions of Maximally Recoverable Local Reconstruction Codes via Function Fields
No full textLocal Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an (n,r,h,a)-LRC, the n codeword symbols are partitioned into r disjoint groups each of which include a local parity checks capable of locally correcting a erasures. The codeword symbols further obey h heavy (global) parity checks. Such a code is maximally recoverable if it can correct all patterns of a erasures per local group plus up to h additional erasures anywhere in the codeword. This property amounts to linear independence of all such subsets of columns of the parity check matrix.
MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that is exponential in h or a, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small r =slant Omega(n^{1-epsilon}), we improve the field size from roughly n^h to n^{epsilon h}. For the case of a=1 (one local parity check), we improve the field size quadratically from r^{h(h+1)} to r^{h floor[(h+1)/2]} for some range of r. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea
Efficiently list-decodable insertion and deletion codes via concatenation
No full textIn this paper, we consider the list decoding property of codes under insertion and deletion errors (insdel for short). Firstly, we analyse the list decodability of random insdel codes. Our result provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of insdel codes along with their efficient encoding and decoding algorithms through concatenation method which provides a Zyablov-type bound for insdel metric codes.Info-communications Media Development Authority (IMDA)National Research Foundation (NRF)Submitted/Accepted versionThe work of Shu Liu was supported in part by the National Natural Science Foundation of China under Grant 11901077, Grant 2019YFB1803102, and Grant 2020YFB1806805; in part by the National Key Research and Development Program of China under Contract IFN2020203; and in part by the Fundamental Research Funds for the Central Universities of China under Grant ZYGX2018KYQD216. The work of Ivan Tjuawinata was supported by the National Research Foundation, Singapore, under its Strategic Capability Research Centres Funding Initiative. The work of Chaoping Xing was supported in part by the National Natural Science Foundation of China under Grant 12031011 and in part by the National Key Research and Development Project under Grant 2020YFA0712300
How Long Can Optimal Locally Repairable Codes Be?
No full textA locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets.
Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case
Leakage-resilient secret sharing with constant share size
No full textIn this work, we consider the leakage-resilience of algebraic-geometric (AG for short) codes based ramp secret sharing schemes extending the analysis on the leakage-resilience of linear threshold secret sharing schemes over prime fields that is done by Benhamouda et al. in the effort to construct linear leakage-resilient secret sharing schemes with constant share size. Since there does not exist any explicit efficient construction of AG codes over prime fields with constant field size, we consider constructions over prime fields with the help of concatenation method and constructions of codes over field extensions. Extending the Fourier analysis done by Benhamouda et al., one can show that concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curious question is whether AG codes over extension fields produce better leakage-resilient secret sharing schemes than the construction based on concatenated AG codes. Such construction provides several advantage compared to the construction over prime fields using concatenation method. It is clear that AG codes over extension fields give secret sharing schemes with a smaller reconstruction threshold for a fixed privacy parameter t. In this work, it is also confirmed that indeed AG codes over extension fields have stronger leakage-resilience under some reasonable assumptions. Furthermore, we also show that AG codes over extension fields may provide strong multiplicative property which may be used in its application to the study of multiparty computation. In contrast, the same cannot be said for constructions based on concatenated AG codes, even when we are considering multiplication friendly embeddings. These advantages strongly motivate the study of secret sharing schemes from AG codes over extension fields. The current paper has two main contributions: (i) we obtain leakage-resilient secret sharing schemes with constant share sizes and unbounded numbers of players. Some of the schemes constructed without the use of concatenation also possesses strong multiplicative property (ii) via Fourier Analysis, we analyze the leakage-resilience of secret sharing schemes from codes over extension fields. This is of its own theoretical interest independent of its application to secret sharing schemes from algebraic geometric codes over extension fields.National Research Foundation (NRF)Submitted/Accepted versionThe work of Ivan Tjuawinata was supported by the National Research Foundation, Singapore, under its Strategic Capability Research Centres Funding Initiative. The work of Chaoping Xing was supported in part by the National Key Research and Development Project under Grant 2021YFE0109900 and Grant 2020YFA0712300 and in part by the Natural Science Foundation of China under Grant 12031011
Explicit construction of q-ary 2-deletion correcting codes with low redundancy
No full textWe consider the problem of efficient construction of q -ary 2-deletion correcting codes with low redundancy. We show that our construction requires less redundancy than any existing efficiently encodable q -ary 2-deletion correcting codes. Precisely speaking, we present an explicit construction of a q -ary 2-deletion correcting code with redundancy 5 log n +10 log log n + 3 log q + O (1) where q is assumed to be a constant with respect to n . Using a minor modification to the original construction, we obtain an efficiently encodable q -ary 2-deletion code that is efficiently list-decodable. Similarly, we show that our construction of list-decodable code requires a smaller redundancy compared to any existing list-decodable codes. To obtain our sketches, we transform a q -ary code-word to a binary string which can then be used as an input to the underlying base binary sketch. This is then complemented with additional q -ary sketches that the original q -ary codeword is required to satisfy. In other words, we build our codes via a binary 2-deletion code as a black-box. Finally we utilize the binary 2-deletion code proposed by Guruswami and Håstad to our construction to obtain the main result of this paper.National Research Foundation (NRF)Submitted/Accepted versionThe work of Shu Liu was supported in part by the National Key R&D Program of China under Grant 2023YFE0123900, 2022YFA1004900, in part by the National Natural Science Foundation of China under Grant 12271084, 12361141818, in part by Young Elite Scientists Sponsorship Program by CAST under Grant 2023QNRC001, in part by the National Key Laboratory of Wireless Communications Foundation under Grant IFN20230107. This research is supported by the National Research Foundation, Singapore under its Strategic Capability Research Centres Funding Initiative. The work of Chaoping Xing was supported in part by the National Natural Science Foundation of China under Grants 1203101
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