17 results
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2. The Subfield Codes of Some [ q + 1, 2, q ] MDS Codes.
- Author
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Heng, Ziling and Ding, Cunsheng
- Subjects
FINITE fields ,HAMMING weight ,LINEAR codes - Abstract
Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Shortened Linear Codes From APN and PN Functions.
- Author
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Xiang, Can, Tang, Chunming, and Ding, Cunsheng
- Subjects
LINEAR codes ,BINARY codes ,CODING theory ,REED-Muller codes ,GENERATING functions ,NONLINEAR functions - Abstract
Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. Approximate Unitary Designs Give Rise to Quantum Channels With Super Additive Classical Holevo Capacity.
- Author
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Nema, Aditya and Sen, Pranab
- Subjects
MONOTONIC functions ,DIFFERENTIABLE functions ,BEHAVIORAL assessment ,ADDITIVES ,QUANTUM mechanics - Abstract
In a breakthrough, Hastings showed that there exist quantum channels whose classical Holevo capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings’ proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate $n^{2/3}$ -design gives rise to a quantum channel with superadditive classical Holevo capacity, where $n$ is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We follow the geometric functional analytic approach of Aubrun, Szarek and Werner in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximate $t$ -design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique for proving concentration of measure for an approximate $t$ -design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Haar random unitaries require at least $\Omega (n^{2})$ random bits in order to describe them with good precision. In contrast, there exist exact $n^{2/3}$ -designs using only $O(n^{2/3} \log n)$ random bits. Thus, our work can be viewed as a partial derandomisation of Hastings’ result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity. Finally we also show that for any $p > 1$ , approximate unitary $n^{1.7}$ -designs give rise to channels violating subadditivity of Rényi $p$ -entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
5. Decoding Reed–Solomon Skew-Differential Codes.
- Author
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Gomez-Torrecillas, Jose, Navarro, Gabriel, and Sanchez-Hernandez, Jose Patricio
- Subjects
REED-Solomon codes ,LINEAR codes ,NONCOMMUTATIVE rings ,DECODING algorithms ,PARITY-check matrix ,LINEAR operators ,POLYNOMIAL rings - Abstract
A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accessible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
6. On the Capacity Enlargement of Gaussian Broadcast Channels With Passive Noisy Feedback.
- Author
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Ravi, Aditya Narayan, Pillai, Sibi Raj B., Prabhakaran, Vinod M., and Wigger, Michele
- Subjects
GAUSSIAN channels ,BROADCAST channels ,RANDOM noise theory ,GAUSSIAN processes ,CHANNEL coding ,PSYCHOLOGICAL feedback - Abstract
It is well known that the capacity region of an average transmit power constrained Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers is enlarged by the presence of causal noiseless feedback. When the noise variances at the receivers are identical, even passive feedback via independent memoryless Gaussian links can lead to a capacity region enlargement. The last fact remains true even when the feedback noise variance is very high, and available only from one of the receivers. While such capacity enlargements are feasible for several other feedback models in the Gaussian BC setting, it is also known that feedback does not change the capacity region for physically degraded broadcast channels. In this paper, we consider a two user GBC with independent noise realizations at the receivers, where the feedback links from the receivers are corrupted by independent additive Gaussian noise processes. We investigate the set of four noise variances, two forward and two feedback, for which no capacity enlargement is possible. A sharp characterization of this region is derived, i.e., any quadruple outside the presented region will lead to a capacity enlargement, whereas quadruples inside will leave the capacity region unchanged. Our results lead to the conclusion that when the forward noise variances are different, too noisy a feedback from one of the receivers alone is not always beneficial for enlarging the capacity region, be it from the stronger user or the weaker one, in sharp contrast to the case of equal forward noise variances. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
7. Three New Constructions of Asymptotically Optimal Periodic Quasi-Complementary Sequence Sets With Small Alphabet Sizes.
- Author
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Luo, Gaojun, Cao, Xiwang, Shi, Minjia, and Helleseth, Tor
- Subjects
HADAMARD codes ,CODE division multiple access ,FINITE fields - Abstract
Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
8. On a Tracial Version of Haemers Bound.
- Author
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Gao, Li, Gribling, Sander, and Li, Yinan
- Subjects
QUANTUM numbers ,THETA functions ,QUANTUM entanglement - Abstract
We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over $\mathbb {C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes’ embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Additive Complementary Dual Codes From Group Characters.
- Author
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Dougherty, Steven T., Sahinkaya, Serap, and Ustun, Deniz
- Subjects
ALGEBRAIC coding theory ,LINEAR codes ,QUANTUM computing ,ABELIAN groups ,BINARY codes ,FINITE groups - Abstract
Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths $n \leq 10$. The obtained codes are either optimal or near optimal according to Bierbrauer et al +. (2009). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. A Combinatorial Interpretation for the Shor-Laflamme Weight Enumerators of CWS Codes.
- Author
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Nemec, Andrew and Klappenecker, Andreas
- Subjects
ERROR-correcting codes ,QUANTUM entanglement ,LINEAR programming - Abstract
We show that one of the Shor-Laflamme weight enumerators of a codeword stabilized quantum code may be interpreted as the distance enumerator of an associated classical code. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. Asymptotic Continuity of Additive Entanglement Measures.
- Subjects
QUANTUM communication ,CONTINUITY ,HILBERT space - Abstract
We study rates of asymptotic transformations between entangled states by local operations and classical communication and a sublinear amount of quantum communication. It is known that additive asymptotically continuous entanglement measures provide upper bounds on the rates that are achievable with asymptotically vanishing error. We show that for transformations between pure states, the optimal rate between any pair of states can be characterized as the infimum of such upper bounds provided by fully additive asymptotically continuous entanglement measures. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
12. Bounds for List-Decoding and List-Recovery of Random Linear Codes.
- Author
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Guruswami, Venkatesan, Li, Ray, Mosheiff, Jonathan, Resch, Nicolas, Silas, Shashwat, and Wootters, Mary
- Subjects
BINARY codes ,CODING theory ,LINEAR codes - Abstract
A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$. It is said to be list-recoverable for input list size $\ell $ if for every sufficiently large subset of at least $L$ codewords, there is a coordinate where the codewords take more than $\ell $ values. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below $q$ is the alphabet size, and $ \varepsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log _{q}(\ell) - \varepsilon $ requires list size $L \ge \ell ^{\Omega (1/ \varepsilon)}$ for list-recovery from input list size $\ell $. (2) A random linear code of rate $1 - h_{q}(p) - \varepsilon $ requires list size $L \ge \left \lfloor{ {h_{q}(p)/ \varepsilon +0.99}}\right \rfloor $ for list-decoding from error fraction $p$. (3) A random binary linear code of rate $1 - h_{2}(p) - \varepsilon $ is list-decodable from average error fraction $p$ with list size with $L \leq \left \lfloor{ {h_{2}(p)/ \varepsilon }}\right \rfloor + 2$. Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset—or some symbol-wise permutation of it—lies in a random linear code with high probability. Our upper bound follows by strengthening a result of (Li, Wootters, 2018). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
13. ℤ₂ℤ₄-Additive Quasi-Cyclic Codes.
- Author
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Shi, Minjia, Li, Shitao, and Sole, Patrick
- Subjects
LINEAR codes ,CYCLIC codes ,BINARY codes ,CHINESE remainder theorem ,CATHODE ray tubes ,LIQUID crystal displays - Abstract
We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representation and exactly enumerated. In particular, we present some illustrative examples of binary optimal linear codes with respect to the Griesmer bound derived from the $\mathbb {Z}_{2} \mathbb {Z}_{4}$ -additive quasi-cyclic codes. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
14. On Error Exponents of Encoder-Assisted Communication Systems.
- Subjects
TELECOMMUNICATION systems ,GAUSSIAN channels ,CHANNEL coding ,EXPONENTS ,ERROR probability ,COMMUNICATION models ,ADDITIVE white Gaussian noise channels - Abstract
We consider a point–to–point communication system, where in addition to the encoder and the decoder, there is a helper that observes non–causally the realization of the noise vector and provides a (lossy) rate– $ {R}_{{ \text { h}}}$ description of it to the encoder ($ {R}_{{ \text { h}}} < \infty $). In continuation to Lapidoth and Marti (2020), who derived the capacity of this model, here our focus is on error exponents. We consider both continuous–alphabet, additive white Gaussian channels and finite–alphabet, modulo–additive channels, and for each one of them, we study the cases of both fixed–rate and variable–rate noise descriptions by the helper. Our main finding is that, as long as the channel–coding rate, R, is below the helper–rate, $ {R}_{{ \text { h}}}$ , the achievable error exponent is unlimited (i.e., it can be made arbitrarily large), and in some of the cases, it is even strictly infinite (i.e., the error probability can be made strictly zero). However, in the range of coding rates $({R}_{{ \text { h}}}, {R}_{{ \text { h}}}+ {C}_{0})$ , C0 being the ordinary channel capacity (without help), the best achievable error exponent is finite and strictly positive, although there is a certain gap between our upper bound (converse bound) and lower bound (achievability) on the highest achievable error exponent. This means that the model of encoder–assisted communication is essentially equivalent to a model, where in addition to the noisy channel between the encoder and decoder, there is also a parallel noiseless bit–pipe of capacity $ {R}_{{ \text { h}}}$. We also extend the scope to the Gaussian multiple access channel (MAC) and characterize the rate sub–region, where the achievable error exponent is unlimited or even infinite. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
15. Rank and Kernel of Additive Generalized Hadamard Codes.
- Author
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Dougherty, Steven T., Rifa, Josep, and Villanueva, Merce
- Subjects
HADAMARD codes ,VECTOR spaces ,HAMMING distance ,HADAMARD matrices - Abstract
A subset of a vector space $\mathbb {F}_{q}^{n}$ is additive if it is a linear space over the field $\mathbb {F}_{p}$ , where $q=p^{e}$ , $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case $e=2$ , it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
16. Entropy and Relative Entropy From Information-Theoretic Principles.
- Author
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Gour, Gilad and Tomamichel, Marco
- Subjects
ENTROPY (Information theory) ,RANDOM variables ,INFORMATION theory ,AXIOMS - Abstract
We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy satisfying these axioms must lie between the Rényi divergences of order 0 and $\infty $. We further show simple conditions for positive definiteness of such relative entropies and a characterisation in terms of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
17. Finding Compositional Inverses of Permutations From the AGW Criterion.
- Author
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Niu, Tailin, Li, Kangquan, Qu, Longjiang, and Wang, Qiang
- Subjects
RESEARCH & development ,FINITE fields ,INFORMATION theory ,POLYNOMIALS ,BIJECTIONS - Abstract
Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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