Perfect Codes Research Articles

On the non-existence of extended 1-perfect codes and MDS codes

We prove the non-existence of extended perfect codes in H(n,q), where q=3,4 and n>q+2 or both n and q are odd. In particular, we prove the non-existence of non-linear (q+2,qq−1,4)q MDS codes when q is odd.

Non-Binary Diameter Perfect Constant-Weight Codes

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some t-intersecting families related to the well-known Erdos-Ko-Rado theorem, are constructed.

Perfect Codes in Cayley Sum Graphs

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.

Screening vs. gevolution: In chase of a perfect cosmological simulation code

We compare two competing relativistic approaches to the N-body simulation of the Universe large-scale structure. To this end, employing the corresponding alternative computer codes (“gevolution” and “screening”), we conduct a series of cosmological simulations in boxes of different sizes and calculate the power spectra of the scalar perturbation Φ, the frame-dragging vector potential B and the difference between scalar modes χ=Φ−Ψ. We demonstrate that the corresponding power spectra are in very good agreement between the compared schemes. For example, the relative difference of the power spectra for Φ is 0.04% maximum. Since the perturbed Einstein equations have much simpler form in the screening approach, the simulation with this code consumes less computational time, saving almost 40% of CPU hours.

On the weight distribution of perfect binary codes

In this paper, we give a new proof of the closed-form formula for the weight distribution of a perfect binary single-error-correcting code.

On tilings of asymmetric limited-magnitude balls

We study whether an asymmetric limited-magnitude ball may tile Zn. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors.A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.

Nonexistence of perfect permutation codes under the Kendall $$\tau $$-metric

In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in \(S_n\), the set of all permutations on n elements, under the Kendall \(\tau \)-metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in \(S_n\), under the Kendall \(\tau \)-metric, for more values of n. Specifically, we present the polynomial representation of the size of a ball in \(S_n\) under the Kendall \(\tau \)-metric for some radius r, and obtain some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect t-error-correcting code in \(S_n\) under the Kendall \(\tau \)-metric for some n and \(t=2,3,4,5,~\text {or}~\frac{5}{8}\left( {\begin{array}{c}n\\ 2\end{array}}\right) < 2t+1\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \).

Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

The perfect codes of commuting zero divisor graph of some matrices of dimension two

The study of graph properties has gathered many attentions in the past years. The graph properties that are commonly studied include the chromatic number, the clique number and the domination number of a finite graph. In this study, a type of graph properties, which is the perfect code is studied. The perfect code is originally used in coding theory, then extended to other fields including graph theory. Hence, in this paper, the perfect code is determined for the commuting zero divisor graphs of some finite rings of matrices. First, the commuting zero divisor graph of the finite rings of matrices is constructed where its vertices are all zero divisors of the ring and two distinct vertices, say x and y, are adjacent if and only if xy = yx = 0. Then, from the vertex set of the graph, the neighborhood elements of the vertices are determined in order to compute the perfect codes of the graph.

Investigating 1-perfect code using Dominating set

In this article, we investigate the relation between dominating sets and 1-perfect codes. We also study perfect colorings of some Johnson graphs in two colors using linear programming problem.

SIMULATION MODEL OF THE SOFTWARE FOR WORKING WITH PERFECT CODES BASED ON HAMMING CODE AND GOLE CODE

This article presents an analytical review of noise-tolerant codes, namely, the perfect codes are considered. Ideal codes can be used to encode and decode information, as they have all the desired properties, namely, resistance to interference and ease of implementation. In conclusion, a simulation model of encoding and decoding the information word is developed.

Perfect codes in some products of graphs

A r -perfect code in a graph G = ( V ( G ), E ( G )) is a subset C of V ( G ) for which the balls of radius r centered at the vertices of C form a partition of V ( G ). In this paper, we study the existence of perfect codes in corona product and generalized hierarchical product of graphs where the cardinality of U is equal to one or two. Also, we give some examples as applications of our results.

Optimal noise estimation from syndrome statistics of quantum codes

Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough. To make this error correction competitive information about the specific noise is required. Traditionally, this information is obtained by benchmarking the device before operation. We address the question of what can be learned from only the measurements done during decoding. Such estimation of noise models was proposed for surface codes, exploiting their special structure, and in the limit of low error rates also for other codes. However, so far it has been unclear under what general conditions noise models can be estimated from the syndrome measurements. In this work, we derive a general condition for identifiability of the error rates. For general stabilizer codes, we prove identifiability under the assumption that the rates are small enough. Without this assumption we prove a result for perfect codes. Finally, we propose a practical estimation method with linear runtime for concatenated codes. We demonstrate that it outperforms other recently proposed methods and that the estimation is optimal in the sense that it reaches the Cram\'{e}r-Rao Bound. Our method paves the way for practical calibration of error corrected quantum devices during operation.

Optimal Anticodes, Diameter Perfect Codes, Chains and Weights

Let ${P}$ be a partial order on $[{n}] = \{1,2,\ldots,{n}\}$ , $\mathbb {F}_{q}^{n}$ be the linear space of ${n}$ -tuples over a finite field $\mathbb {F}_{q}$ and ${w}$ be a weight on $\mathbb {F}_{q}$ . In this paper, we consider metrics on $\mathbb {F}_{q}^{n}$ induced by chain orders ${P}$ over $[{n}]$ and weights ${w}$ over $\mathbb {F}_{q}$ , and we determine the cardinality of all optimal anticodes and completely classify them. Moreover, we determine all diameter perfect codes for a set of relevant instances on the aforementioned metric spaces.

Perfect codes in Euclidean lattices

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We discuss the connection between a discrete tiling of a lattice and a continuous tilling of the n-dimensional space and equivalent characterizations of discrete tilings. We generalize bounds on the radius of perfect codes in a generic lattice, which were previously known for the cubic lattice, and we provide some new bounds. The new bounds are based on the packing and the covering densities and on the covering radius of the ambient lattice. An algorithm is presented for the search of perfect codes which is used to derive all perfect codes for a collection of ambient lattices in dimensions two and three. In contrast to the cubic lattice, these case studies show that by considering general ambient lattices one can find rich sets of perfect codes.

Two-wavelength phase-sensitive OTDR sensor using perfect periodic correlation codes for measurement range enhancement, noise reduction and fading compensation.

We demonstrate a two-wavelength differential-phase-measuring OTDR sensor that uses perfect periodic correlation phase codes to enhance the measurement performance. The two-wavelength technique extends the measurement range of OTDR sensors by synthesizing a virtual longer-wavelength measurement from two simultaneous measurements of phase using different lasers. This increases the range free from phase unwrapping errors. However, we find that the application of this technique greatly increases the relative measurement noise. To compensate for this issue, we introduce the use of optical pulse compression using perfect periodic correlation phase codes to increase the measurement signal-to-noise ratio and also to facilitate the simultaneous compensation of Rayleigh and polarization fading. In addition, we apply a method to further reduce the relative noise that is added to the two-wavelength measurement by using the synthetic wavelength measurement to unwrap the differential phase measured with a single wavelength. All this is highlighted in a 1-km sensing link in which up to 20-cm spatial resolution and 12.6 pϵ/Hz strain sensitivity are demonstrated as well as a 67-fold enhancement in measurement range compared with the use of the conventional single-wavelength method.

Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits.

Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of n}{}57.1(3)% while with a high fidelity of n}{}98.6(1)% in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of n}{}97.2(2)% within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of n}{}74.5(6)%, in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits.

On the Generalized Concatenated Construction for Codes in $${L_1}$$ and Lee Metrics

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L1 and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2m we construct codes over $$\mathbb{Z}_4$$ with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2m+1 (with code distance 4). We construct codes over $$\mathbb{Z}_4$$ of length n with Lee distance n which under the Gray mapping yield Hadamard matrices of order 2n (under the additional condition that an Hadamard matrix of order n exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.

STOLARSKY'S INVARIANCE PRINCIPLE FOR FINITE METRIC SPACES

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.

MDS and I-Perfect Codes in Pomset Metric

Recently pomset metric was introduced to accommodate Lee metric, thereby generalizing poset metrics for codes over $\mathbb {Z}_{{m}}$ . This work mainly studies results on Maximum distance separable (MDS) pomset codes and perfect codes over $\mathbb {Z}_{{m}}$ . If I is an ideal in a poset, the concept of I -perfect codes that was introduced to study MDS poset codes is now extended to the study of MDS pomset codes as well. But, unlike the ideals in posets, here the ideals in pomsets are considered as those with full count and those with partial count. In fact, the I -balls are no more linear when the ideal I is with partial count. This makes the investigation of the relationship of I -perfect codes with MDS pomset codes little different from that with MDS poset codes. We thoroughly study the I -balls where I is with partial count. Singleton type bound is established for codes with pomset metric and the connection of MDS codes with I -perfect codes is investigated upon. Then, we prove the duality theorem for codes with chain pomset. Finally, we determine the weight distribution of MDS pomset codes when the pomset is a chain.