Constant Weight Codes Research Articles

Combinatorial Limitations of Average-Radius List-Decoding

We study certain combinatorial aspects of listdecoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (log(1/γ))) for the list size needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 - h(p) - γ [here p E (0, 1/2) and γ > 0]. Our main result is that we prove that in any binary code C ⊆ (0, 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> of rate 1 - h(p) - γ, there must exist a set l ⊂ C of p(1/√γ) codewords such that the average distance of the points in L from their centroid is at most pn. In other words, there must exist Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (1/√γ) codewords with low average radius. The standard notion of list decoding corresponds to working with the maximum distance of a collection of codewords from a center instead of average distance. The average radius form is in itself quite natural; for instance, the classical Johnson bound in fact implies average-radius list-decodability. The remaining results concern the standard notion of list-decoding, and help clarify the current state of affairs regarding combinatorial bounds for list-decoding as follows. First, we give a short simple proof, over all fixed alphabets, of the above-mentioned Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (log(1/γ)) lower bound. Earlier, this bound followed from a complicated, more general result of Blinovsky. Second, we show that one cannot improve the Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (log(1/γ)) lower bound via techniques based on identifying the zero-rate regime for list-decoding of constantweight codes [this is a typical approach for negative results in coding theory, including the Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (log(1/γ)) list-size lower bound]. On a positive note, our Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (1/√γ) lower bound for average radius list-decoding circumvents this barrier. Third, we exhibit a reverse connection between the existence of constant-weight and general codes for list-decoding, showing that the best possible list-size, as a function of the gap γ of the rate to the capacity limit, is the same up to constant factors for both constant-weight codes (with weight bounded away from p) and general codes. Fourth, we give simple second moment-based proofs that w.h.p. a list-size of Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (1/γ) is needed for list-decoding random codes from errors as well as erasures. For random linear codes, the corresponding list-size bounds are Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (1/γ) for errors and expΩ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (log(1/γ)) for erasures.

2-D Non-Isolated Pixel 6/8 Modulation Code

In this paper, we introduce a 2-D non-isolated pixel modulation code to mitigate the problem of 2-D inter-symbol interference, the major obstacle of holographic data storage systems. The proposed modulation code does not allow a pixel to be isolated from its adjacent bits. We compare the bit error rate (BER) performance between the proposed and the conventional 6/8 balanced modulation codes. The proposed code shows better BER performance than that of the conventional 6/8 balanced code in a poor channel environment, such as large blur and misalignments.

Bent vectorial functions and linear codes from o-polynomials

The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and $$q$$q-ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. We show that the hyperovals of $$PG_{2}(2^m)$$PG2(2m) from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of $$2^r$$2r-ary ($$r$$r being a divisor of $$m$$m) simplex linear codes (whose duals are the perfect $$2^r$$2r-ary Hamming codes) over an extension field $${\mathbb F}_{2^{r}}$$F2r of $${\mathbb F}_{2^{}}$$F2. The following diagram gives an indication of the main topics and interconnections arising in this paper. [Figure not available: see fulltext.]

Completely reducible super-simple designs with block size five and index two

Complete reducible super-simple (CRSS) designs are closely related to $$q$$q-ary constant weight codes. A $$(v,k,\lambda )$$(v,k,?)-CRSS design is just an optimal $$(v,2(k-1),k)_{\lambda +1}$$(v,2(k-1),k)?+1 code. In this paper, we mainly investigate the existence of a $$(v,5,2)$$(v,5,2)-CRSS design and show that such a design exists if and only if $$v\equiv 1,5\pmod {20}$$v?1,5(mod20) and $$v\ge 21$$v?21, except possibly when $$v = 25$$v=25. Using this result, we determine the maximum size of an $$(n,8,5)_3$$(n,8,5)3 code for all $$n\equiv 0,1,4,5 \pmod {20}$$n?0,1,4,5(mod20) with the only length $$n=25$$n=25 unsettled. In addition, we also construct super-simple $$(v,5,3)$$(v,5,3)-BIBDs for $$v=45,65$$v=45,65.

Fidelity Lower Bounds for Stabilizer and CSS Quantum Codes

In this paper, we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared with stabilizer codes.

Constant Weight Codes: An Approach Based on Knuth's Balancing Method

In this article, we study properties and algorithms for constructing sets of constant weight codewords with bipolar symbols, where the sum of the symbols is a constant q, q\neq 0. We show various code constructions that extend Knuth's balancing vector scheme, q=0, to the case where q>0. We compute the redundancy of the new coding methods. Finally, we generalize the proposed methods to encoding of imbalanced arrays in two or more dimensions.

Low-weight error-prevention codes for electromagnetic nanonetworks in the Terahertz Band

In this paper, a novel error control strategy for electromagnetic nanonetworks, based on the utilization of low-weight channel codes and aimed at the prevention of channel errors, is proposed. In particular, it is first analytically shown that both the molecular absorption noise and the multi-user interference in nanonetworks can be mitigated by reducing the channel code weight, which results into a lower channel error probability. Then, the relation between the channel code weight and the code word length is analyzed for the case of utilizing constant weight codes. Finally, the performance of the proposed strategy is analytically and numerically investigated in terms of the achievable information rate after coding and the Codeword Error Rate (CER). Two different receiver architectures are considered, namely, an ideal soft-receiver and a hard receiver. An accurate Terahertz Band channel model and novel stochastic models for the molecular absorption noise and the multi-user interference, validated with COMSOL, are utilized. The results show that low-weight channel codes can be used to reduce the CER without compromising the achievable information rate or even increasing it, especially for the hard-receiver architecture. Moreover, it is shown that there is an optimal code weight, for which the information rate is maximized.

The t-wise intersection of relative two-weight codes

The t-wise intersection of constant-weight codes are computed. Based on the above result, the t-wise intersection of relative two-weight codes are determined by using the finite geometric structure of relative two-weight codes.

Mars surface thermal inertia and heterogeneities from OMEGA/MEX

The thermophysical structure of the martian surface is the result of various processes that have shaped the martian surface through time. Previous dedicated heliosynchronous measurements of the thermal infrared (IR) flux of the martian surface have revealed the diversity of martian surface thermal properties, as well as its complexity linked to the heterogeneous nature of terrains. We present the first retrieval of thermophysical properties of the martian surface using near-infrared (NIR) Observatoire pour la Minéralogie, l’Eau, les Glaces et l’Activité (OMEGA) onboard Mars Express (MEX) thermal measurements from 5 to 5.1μm. MEX orbit around Mars is elliptical and therefore OMEGA has performed surface temperature measurements at various local times and seasons over more than 4 full martian years. We have developed a method to exploit these unprecedented measurements using a one-dimensional energy balance code derived from a Global Climate Model that allows retrieval of the thermal properties of the martian surface using OMEGA data. Regional maps of the thermal inertia at a resolution up to 32 pixels per degree and a global map at 4 pixels per degree are presented. OMEGA-derived thermal inertia values agree with previous mappings by the Thermal Emission Spectrometer (TES) onboard Mars Global Surveyor (MGS) and Thermal Emission Imaging Spectrometer (THEMIS) onboard Mars Odyssey and highlight the key role of dust for the thermal behavior of the martian surface. OMEGA directly reveals for the first time some diurnal variations of apparent TI attributable to surface heterogeneities at macroscopic scale and enables to quantify these heterogeneities. In Nili Patera and Tharsis, local surface heterogeneities are modeled with layering and horizontal admixture of divergent slopes respectively.

New inequalities for $$q$$ q -ary constant-weight codes

Using double counting, we prove Delsarte inequalities for $$q$$ q -ary codes and their improvements. Applying the same technique to $$q$$ q -ary constant-weight codes, we obtain new inequalities for $$q$$ q -ary constant-weight codes.

Hanani triple packings and optimal $$q$$ q -ary codes of constant weight three

The exact sizes of optimal $$q$$ q -ary codes of length $$n$$ n , constant weight $$w$$ w and distance $$d=2w-1$$ d = 2 w - 1 have only been determined for $$q\in \{2,3\}$$ q � { 2 , 3 } , and for $$w|(q-1)n$$ w | ( q - 1 ) n and $$n$$ n sufficiently large. We completely determine the exact size of optimal $$q$$ q -ary codes of constant weight three and minimum distance five for all $$q$$ q by establishing a connection with Hanani triple packings, and settling their existence.

Bounds on the rate of disjunctive codes

A binary code is said to be a disjunctive (s, l) cover-free code if it is an incidence matrix of a family of sets where the intersection of any l sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L ? 1 other sets of this family. For L = l = 1, both definitions coincide, and the corresponding binary code is called a disjunctive s-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive (s, l) cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as s ? ?, with an arbitrary fixed l ? 1, to the limit 2e?2 = 0.271 ... In the classical case of l = 1, this means that the upper bound on the rate of disjunctive s-codes constructed in 1982 by D'yachkov and Rykov is asymptotically attained up to a constant factor a, 2e?2 ≤ a ≤ 1.

On relative constant-weight codes

In this paper, relative two-weight and three-weight codes are studied, which are both called relative constant-weight codes. A geometric approach is introduced to construct and characterize relativ...

저밀도 ON 픽셀 2차원 6/8 변조부호

Since holographic data storages read and write information on a volume and the information is processed per page, it has the advantage of high recording density and data transfer rate. However, there are two major drawbacks like 2-dimensional intersymbol interference and interpage interference as the density between pixels increases. Furthermore, a bright page that contains many ON pixels influences the reliable detection of the neighboring pages, which causes the less number of pages stored in the storage volume. We propose a sparse-ON pixel two-dimensional modulation code with the code rate 6/8 for increasing the number of pages stored in the volume. The proposed code is compared to conventional 6/8 balanced code, and it shows similar or a little bit better performance than that of the balanced code. Therefore, the proposed code can increase the recording capacity without loss of the performance.

Parsing a Sequence of Qubits

We develop a theoretical framework for frame synchronization, also known as block synchronization, in the quantum domain which makes it possible to attach classical and quantum metadata to quantum information over a noisy channel even when the information source and sink are frame-wise asynchronous. This eliminates the need of frame synchronization at the hardware level and allows for parsing qubit sequences during quantum information processing. Our framework exploits binary constant-weight codes that are self-synchronizing. Possible applications may include asynchronous quantum communication such as a self-synchronizing quantum network where one can hop into the channel at any time, catch the next coming quantum information with a label indicating the sender, and reply by routing her quantum information with control qubits for quantum switches all without assuming prior frame synchronization between users.

Tabular construction of balanced codes

Symmetries in the encoding of balanced sequences are exploited to simplify a tabular approach for constructing balanced codes, and it is shown that with appropriate assignment of source word probabilities, these symmetries ensure the absence of discrete components in the spectrum of the encoded signal. He is further shown that assigning appropriate priorities to codewords during table construction results in superior low-frequency performance, and the use of these techniques is demonstrated in the construction of a balanced QPSK code.

Optimal Quaternary Constant-Weight Codes With Weight Four and Distance Five

Constant-weight codes play an important role in coding theory. The problem of determining the sizes for optimal quaternary constant-weight codes with length <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> , weight 4, and minimum Hamming distance 5 ( <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(n,5,4)_{4}$</tex></formula> codes) has been investigated in several papers. Although some constructions and several infinite families for such codes with length <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n\equiv 0,1\pmod4$</tex></formula> have been given, the problem is still far from complete. In this paper, we determine the size of an optimal <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(n,5,4)_{4}$</tex></formula> code for each integer <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n\geq 4$</tex></formula> leaving 55 lengths unsolved. Especially, for length <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n\equiv 0,1\pmod 4$</tex></formula> , the existence problem of the equivalent combinatorial object, namely the generalized Steiner system, is solved leaving only seven values undetermined.

Estimates on the Size of Symbol Weight Codes

The study of codes for powerline communications has garnered much interest over the past decade. Various types of codes such as permutation codes, frequency permutation arrays, and constant composition codes have been proposed over the years. In this paper, we study a type of code called bounded symbol weight codes which was first introduced by Versfeld in 2005, and a related family of codes that we term constant symbol weight codes. We provide new upper and lower bounds on the size of bounded symbol weight and constant symbol weight codes. We also give direct and recursive constructions of codes for certain parameters.

Some Constant Weight Codes from Primitive Permutation Groups

In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64. Andries Brouwer maintains web pages with tables of the best known constant weight codes of these lengths. In many cases the codes have more codewords than the best code in the literature, and are not particularly easy to improve. Many of the codes are constructed using a specified permutation group as automorphism group. The groups used include cyclic, quasi-cyclic, affine general linear groups etc. sometimes with fixed points. The precise rationale for the choice of groups is not clear.In this paper the choice of groups is made systematic by the use of the classification of primitive permutation groups. Together with several improved techniques for finding a maximum clique, this has led to the construction of 39 improved constant weight codes.

Room squares with super-simple property

Constant-composition codes are a special type of constant-weight codes and have attracted recent interest due to their numerous applications. In a recent work, the authors found that an optimal (n, 5, [2, 1, 1])4-code of constant-composition can be obtained from a Room square of side n with super-simple property. In this paper, we study the existence problem of super-simple Room squares. The problem is solved leaving only two minimal possible n undetermined.