Optimal Conflict-avoiding Codes Research Articles

Certain diagonal equations and conflict-avoiding codes of prime lengths

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g2Xℓ+gYℓ+1=0 over the finite field Fp for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field Fq of Fp. We show that for q greater than a lower bound of the order of magnitude O(ℓ2) there exists a generator g of Fq× such that the equation in question is solvable over Fq. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

On Tight Optimal Conflict-Avoiding Codes for 3, 4, 5 and 6 Active Users

Abstract We classify tight optimal conflict-avoiding codes of weights 3, 4, 5 and 6 and given small lengths.

Splitter Sets and $k$ -Radius Sequences

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict-avoiding codes. The study of $k$ -radius sequences was motivated by some problems occurring in large data transfer. It is observed that the existence of splitter sets yields $k$ -radius sequences of short length. In this paper, we obtain several new results contributing to splitter sets and $k$ -radius sequences. We give some new constructions of perfect splitter sets, as well as some nonexistence results on them. As a byproduct, we obtain some new results on optimal conflict-avoiding codes. Furthermore, we provide several explicit constructions of short $k$ -radius sequences for certain values of $n$ , by establishing the existence of $k$ -additive sequences. In particular, we show that for any fixed $k$ , there exist infinitely many values of $n$ such that $f_{k}(n) =\frac {n^{2}}{2k}+O(n)$ , where $f_{k} (n)$ denotes the shortest length of an $n$ -ary $k$ -radius sequence. This result partially affirms a conjecture posed by Bondy, Lonc, and Rzązewski.

Classification of optimal conflict-avoiding codes of weights 6 and 7

A conflict-avoiding code is used to guarantee that each transmitting user can send at least one packet successfully within a fixed period of time, provided that at most k out of M potential users are active simultaneously in a multiple-access collision channel. The number of codewords in a conflict-avoiding code determines the number M of potential users that can be supported in a system. That is why codes of the maximum cardinality for given parameters (optimal codes) are of interest. In this paper we determine the values of the maximum cardinality and classify up to multiplier equivalence optimal conflict-avoiding codes for 6 and 7 active users and given small lengths.

A new series of optimal tight conflict-avoiding codes of weight [formula omitted

In this article, a construction of an optimal tight conflict-avoiding code of length 3dpe and weight 3 is shown for d≡1(mod3), e∈N and a prime p≡3(mod8) with p≠3, assuming that p is a non-Wieferich prime if e≥2. This is a new series of optimal conflict-avoiding code for which the number of codewords can be exactly determined.

Optimal conflict-avoiding codes for 3, 4 and 5 active users

Conflict-avoiding codes are used in multiple-access collision channels without feedback. The number of codewords in a conflict-avoiding code is the number of potential users of the channel. That is why codes with maximum cardinality (optimal codes) for given parameters are of interest. In this paper we classify, up to multiplier equivalence, all optimal conflict-avoiding codes of weights 3, 4, and 5 and given small lengths. We also determine some previously unknown values of the maximum cardinality of conflict-avoiding codes of weights 4 and 5.

Optimal strongly conflict-avoiding codes of even length and weight three

Strongly conflict-avoiding codes (SCACs) are employed in a slot-asynchronous multiple-access collision channel without feedback to guarantee that each active user can send at least one packet successfully in the worst case within a fixed period of time. Assume all users are assigned distinct codewords, the number of codewords in an SCAC is equal to the number of potential users that can be supported. SCACs have different combinatorial structure compared with conflict-avoiding codes (CACs) due to additional collisions incurred by partially overlapped transmissions. In this paper, we establish upper bounds on the size of SCACs of even length and weight three. Furthermore, it is shown that some optimal CACs can be used to construct optimal SCACs of weight three.

Optimal equi-difference conflict-avoiding codes of weight four

A conflict-avoiding code (CAC) of length $$n$$n and weight $$w$$w is defined as a family $${\mathcal C}$$C of $$w$$w-subsets (called codewords) of $${\mathbb {Z}}_n$$Zn, the ring of residues modulo $$n$$n, such that $$\Delta (C) \cap \Delta (C') = \emptyset $$Δ(C)?Δ(C?)=? for any $$C, C' \in {\mathcal C}$$C,C??C, where $$\Delta (C) = \{ j-i \pmod {n} : i, j \in C, i \ne j\}$$Δ(C)={j-i(modn):i,j?C,i?j}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is called an equi-difference code if every codeword $$C \in {\mathcal C}$$C?C has the form $$\{ 0, i, 2i, \ldots , (w-1) i \}$${0,i,2i,?,(w-1)i}. A code $${\mathcal C}$$C in CACs of length $$n$$n and weight $$w$$w is said to be optimal if $${\mathcal C}$$C has the maximum number of codewords. In this article, we investigate sizes and constructions of optimal codes in equi-difference CACs of weight four by using properly defined directed graphs. As a consequence, several series of infinite number of optimal equi-difference CACs are also provided.

Weighted maximum matchings and optimal equi-difference conflict-avoiding codes

A conflict-avoiding code (CAC) $$\mathcal {C}$$C of length $$n$$n and weight $$k$$k is a collection of $$k$$k-subsets of $$\mathbb {Z}_n$$Zn such that $$\varDelta (x)\cap \varDelta (y)=\emptyset $$Δ(x)?Δ(y)=? for any $$x,y\in \mathcal {C}$$x,y?C and $$x\ne y$$x?y, where $$\varDelta (x)=\{a-b:\, a,b\in x, a\ne b\}$$Δ(x)={a-b:a,b?x,a?b}. Let $$\text {CAC}(n,k)$$CAC(n,k) denote the class of all CACs of length $$n$$n and weight $$k$$k. A CAC $$\mathcal {C}\in \text {CAC}(n,k)$$C?CAC(n,k) is said to be equi-difference if any codeword $$x\in \mathcal {C}$$x?C has the form $$\{ 0,i,2i,\ldots , (k-1)i \}$${0,i,2i,?,(k-1)i}. A CAC with maximum size is called optimal. In this paper we propose a graphical characterization of an equi-difference CAC, and then provide an infinite number of optimal equi-difference CACs for weight four.

Optimal equi-difference conflict-avoiding codes of odd length and weight three

A conflict-avoiding code (CAC) is known as a protocol sequence for transmitting data packets over a collision channel without feedback. The study of CACs has been focused on determining the size of an optimal code, i.e., the maximum size of a code, and in the past few years it has been settled by several researchers for even length and weight 3 together with constructions. As for odd length, a necessary and sufficient condition for the existence of a ‘tight equi-difference’ CAC of weight 3 can be found in Momihara (2007), but the condition is fairly complex and thus only a few explicit series of code lengths are known. Recently, Fu et al. (2013) restated the condition given by Momihara (2007) in a different way, which requires to examine the multiplicative suborder of 2 modulo p for each prime factor p of m. Meanwhile, Ma et al. (2013) presented constructions of an optimal equi-difference CAC and an optimal tight CAC of odd prime length p and weight 3, and formulated the sizes of such optimal codes. However, for their formulae to have practical meaning, the number of cosets of −(2)p∪(2)p still needs to be determined, where (2)p is the multiplicative subgroup of Zp⁎ with generator 2. Moreover, their construction of an optimal tight CAC imposes a certain condition. This implies that even restricting ourselves to odd prime length, to provide a series of odd code length for which the maximum size of a CAC of weight 3 can be determined is a demanding problem.In this article, we will give some explicit series of tight/optimal equi-difference CACs of odd length and weight 3 by revisiting some properties of multiplicative order of a unit in the ring of residues modulo m and cyclotomic polynomials.

New optimal constructions of conflict-avoiding codes of odd length and weight 3

Conflict-avoiding codes (CACs) have played an important role in multiple-access collision channel without feedback. The size of a CAC is the number of codewords which equals the number of potential users that can be supported in the system. A CAC with maximal code size is said to be optimal. The use of an optimal CAC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, the maximal sizes of both equidifference and non equidifference CACs of odd prime length and weight 3 are obtained. Meanwhile, the optimal constructions of both equidifference and non equidifference CACs are presented. The numbers of equidifference and non equidifference codewords in an optimal code are also obtained. Furthermore, a new modified recursive construction of CACs for any odd length is shown. Non equidifference codes can be constructed in this method.

Optimal conflict-avoiding codes of odd length and weight three

A conflict-avoiding code (CAC) $${\mathcal{C}}$$ of length n and weight k is a collection of k-subsets of $${\mathbb{Z}_{n}}$$ such that $${\Delta (x) \cap \Delta (y) = \emptyset}$$ for any $${x, y \in \mathcal{C}}$$ , $${x\neq y}$$ , where $${\Delta (x) = \{a - b:\,a, b \in x, a \neq b\}}$$ . Let CAC(n, k) denote the class of all CACs of length n and weight k. A CAC with maximum size is called optimal. In this paper, we study the constructions of optimal CACs for the case when n is odd and k = 3.

Errata to “Optimal Conflict-Avoiding Codes of Even Length and Weight 3” [Nov 10 5747-5756

In the above titled paper (ibid., vol. 56, pp. 5747-5756, Nov. 2010), the following corrections are necessary.

Optimal Conflict-Avoiding Codes of Even Length and Weight 3

Direct constructions for optimal conflict-avoiding codes of length $n \equiv 4 \pmod{8}$ and weight 3 are provided by bringing in a new concept called an extended odd sequence. Constructions for those odd sequences are also given in this paper. As a consequence, with previously known results, the spectrum of the size of optimal conflict-avoiding codes of even length and weight 3 is completely settled.

Optimal conflict-avoiding codes of length n ≡ 0 (mod 16) and weight 3

A conflict-avoiding code of length n and weight k is defined as a set $${C \subseteq \{0,1\}^n}$$ of binary vectors, called codewords, all of Hamming weight k such that the distance of arbitrary cyclic shifts of two distinct codewords in C is at least 2k−2. In this paper, we obtain direct constructions for optimal conflict-avoiding codes of length n = 16m and weight 3 for any m by utilizing Skolem type sequences. We also show that for the case n = 16m + 8 Skolem type sequences can give more concise constructions than the ones obtained earlier by Jimbo et al.

Constant Weight Conflict-Avoiding Codes

A conflict-avoiding code (CAC) $C$ of length $n$ with weight $k$ is a family of binary sequences of length $n$ and weight $k$ satisfying $\sum_{0\le t\le n-1}x_{it}x_{j,t+s}\le \lambda$ for any distinct codewords $x_i=(x_{i0},x_{i1},\ldots,x_{i,n-1})$ and $x_j=(x_{j0},x_{j1},\ldots,x_{j,n-1})$ in $C$ and for any integer $s$, where the subscripts are taken modulo $n$. A CAC with maximum code size for given $n$ and $k$ is said to be optimal. A CAC has been studied for sending messages correctly through a multiple-access channel. The use of an optimal CAC enables the largest possible number of potential users to transmit information efficiently and reliably. In this paper, the case $\lambda=1$ is treated, and various direct and recursive constructions of optimal CACs for weight $k=4$ and $5$ are obtained by providing constructions of CACs for general weight $k$. In particular, the maximum code size of CACs satisfying certain sufficient conditions is determined through number theoretical and combinatorial approaches.

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of $${\mathbb{Z}}_n$$ such that $$\Delta(x) \cap \Delta(y) = \emptyset$$ holds for any $$x,y\in C$$ , $$x\not= y$$ , where $$\Delta(x)=\{j-i\,|\, i,j\in x, i\not= j\}$$ . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if $$\bigcup_{x\in C}\Delta(x)={\mathbb{Z}}_n\setminus \{0\}$$ holds and any codeword $$x\in C$$ has the form $$\{0,i,2i,\ldots,(k-1)i\}$$ . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n's admitting the condition through a number theoretical approach.

On Conflict-Avoiding Codes of Length $n=4m$ for Three Active Users

New improved upper and lower bounds on the maximum size of a symmetric or arbitrary conflict-avoiding code of length n = 4 m for three active users are proved. Furthermore, direct constructions for optimal conflict-avoiding codes of length n = 4 m and m equiv 2 (mod 4) for three active users are provided.