A permutation array(or code) of length $n$ and distance $d$, denoted by $(n,d)$ PA, is a set of permutations $C$ from some fixed set of $n$ elements such that the Hamming distance between distinct members $\mathbf{x},\mathbf{y}\in C$ is at least $d$. Let $P(n,d)$ denote the maximum size of an $(n,d)$ PA. This correspondence focuses on the lower bound on $P(n,d)$. First we give three improvements over the Gilbert-Varshamov lower bounds on $P(n,d)$ by applying the graph theorem framework presented by Jiang and Vardy. Next we show another two new improved bounds by considering the covered balls intersections. Finally some new lower bounds for certain values of $n$ and $d$ are given.
Permutation arrays under the Chebyshev metric have been considered for error correction in noisy channels. Let $P(n,d)$ denote the maximum size of any array of permutations on $n$ symbols with pairwise Chebyshev distance $d$. We give new techniques and improved upper and lower bounds on $P(n,d)$, including a precise formula for $P(n,2)$.
We give new lower bounds for $M(n,d)$, for various positive integers $n$ and $d$ with $n>d$, where $M(n,d)$ is the largest number of permutations on $n$ symbols with pairwise Hamming distance at least $d$. Large sets of permutations on $n$ symbols with pairwise Hamming distance $d$ is a necessary component of constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, {\em partition and extension}, is universally applicable to constructing such sets for all $n$ and all $d$, $d<n$. We describe three new techniques, {\em sequential part…
Let $M(n,d)$ be the maximum size of a permutation array on $n$ symbols with pairwise Hamming distance at least $d$. Some permutation arrays can be constructed using blocks of certain type [2] called product blocks in this paper. We study the problem of designing $(q,k)$-product blocks with a fixed block size $k$.
A permutation array $A$ is a set of permutations on a finite set $Ω$, say of size $n$. Given distinct permutations $π, σ\in Ω$, we let $hd(π, σ) = |\{ x\in Ω: π(x) \ne σ(x) \}|$, called the Hamming distance between $π$ and $σ$. Now let $hd(A) =$ min$\{ hd(π, σ): π, σ\in A \}$. For positive integers $n$ and $d$ with $d\le n$, we let $M(n,d)$ be the maximum number of permutations in any array $A$ satisfying $hd(A) \geq d$. There is an extensive literature on the function $M(n,d)$, motivated in part by suggested applications to error correcting codes for message transmission over power lines. A b…
We consider rational functions of the form $V(x)/U(x)$, where both $V(x)$ and $U(x)$ are polynomials over the finite field $\mathbb{F}_q$. Polynomials that permute the elements of a field, called {\it permutation polynomials ($PPs$)}, have been the subject of research for decades. Let ${\mathcal P}^1(\mathbb{F}_q)$ denote $\mathbb{Z}_q \cup \{\infty\}$. If the rational function, $V(x)/U(x)$, permutes the elements of ${\mathcal P}^1(\mathbb{F}_q)$, it is called a {\em permutation rational function (PRf)}. Let $N_d(q)$ denote the number of PPs of degree $d$ over $\mathbb{F}_q$, and let $N_{v,u}(…
In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[n,k,d]_q$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree $n$, having minimum distance at least $d$ and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and d…
Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing tight upper and lower bounds on multipermutation Ulam sphere sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.
Polar coding solutions demonstrate excellent performance under the list decoding that is challenging to implement in hardware due to the path sorting operations. As a potential solution to this problem, permutation decoding recently became a hot research topic. However, it imposes more constraints on the code structure. In this paper, we study the structural properties of Arikan's polar codes. It is known that they are invariant under lower-triangular affine permutations among others. However, those permutations are not useful in the context of permutation decoding. We show that, unfortunately…
Let $d$ be a positive integer and $x$ a real number. Let $A_{d, x}$ be a $d\times 2d$ matrix with its entries $$ a_{i,j}=\left\{ \begin{array}{ll} x\ \ & \mbox{for} \ 1\leqslant j\leqslant d+1-i, 1\ \ & \mbox{for} \ d+2-i\leqslant j\leqslant d+i, 0\ \ & \mbox{for} \ d+1+i\leqslant j\leqslant 2d. \end{array} \right. $$ Further, let $R_d$ be a set of sequences of integers as follows: $$R_d=\{(ρ_1, ρ_2,\ldots, ρ_d)|1\leqslant ρ_i\leqslant d+i, 1\leqslant i \leqslant d,\ \mbox{and}\ ρ_r\neq ρ_s\ \mbox{for}\ r\neq s\}.$$ and define $$Ω_d(x)=\sum_{ρ\in R_d}a_{1,ρ_1}a_{2, ρ_2}\ldots a_{d,ρ_d}.$$ In o…
The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with \textit{locality}, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. This paper focuses on the combinatorial approach. We provide upper and lower bounds fo…
Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$, $A_2(24,4) \ge 327680$, $A_2(24,10) \ge 136$, and $A_2(25,6) \ge 17920$. The new lower bounds are a result of a systematic computer search over transitive permutation groups.
We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = ρ\cdot (n\!-\!1)$, $ρ\in [0,1]$, i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most $0.029$ bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.
Accessing the data in the failed disk (degraded read) with low latency is crucial for an erasure-coded storage system. In this work, the maximum distance separable (MDS) array code with the property of degraded-read friendly (DRF) is discussed. For the DRF MDS array code with 2 redundant nodes and the sub-packetization level of 2, the lower bound of its access bandwidth is derived.
Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall's $τ$-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert-Varshamov bound asymptotically by a factor $n$, when the minimum …
We show that the underlying permutation of ChaCha20 stream cipher does not behave as a random permutation for up to 17 rounds with respect to rotational cryptanalysis. In particular, we derive a lower and an upper bound for the rotational probability through ChaCha quarter round, we show how to extend the bound to a full round and then to the full permutation. The obtained bounds show that the probability to find what we call a parallel rotational collision is, for example, less than $2^{-488}$ for 17 rounds of ChaCha permutation, while for a random permutation of the same input size, this pro…
