Conflict-avoiding Codes Research Articles

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Conflict-Avoiding Codes of Prime Lengths and Cyclotomic Numbers

Corrections to “Multichannel Conflict-Avoiding Codes of Weights Three and Four”

Corrections to “Multichannel Conflict-Avoiding Codes of Weights Three and Four”

Constructions for Multichannel Conflict-Avoiding Codes With AM-OPPTS Restriction

Constructions for Multichannel Conflict-Avoiding Codes With AM-OPPTS Restriction

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.

Multichannel Conflict-Avoiding Codes of Weights Three and Four

Conflict-avoiding codes (CACs) were introduced by Levenshtein as a single-channel transmission scheme for a multiple-access collision channel without feedback. When the number of simultaneously active source nodes is less than or equal to the weight of a CAC, it is able to provide a hard guarantee that each active source node transmits at least one packet successfully within a fixed time duration, no matter what the relative time offsets between the source nodes are. In this article, we extend CACs to multichannel CACs for providing such a hard guarantee over multiple orthogonal channels. Upper bounds on the number of codewords for multichannel CACs of weights three and four are derived, and constructions that are optimal with respect to these bounds are presented.

Classification of Strongly Conflict-Avoiding Codes

Strongly conflict-avoiding codes (SCACs) for slot-asynchronous multiple-access collision channels without feedback are considered. Codes with maximum size (optimal codes) for given parameters are of interest. In this letter, we present classification results about optimal SCACs of weights 3, 4, and 5 and given small lengths. We also determine some previously unknown values of the size of optimal SCACs with these parameters.

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.

Strongly Conflict-Avoiding Codes with Weight Three

Strongly conflict-avoiding codes are used in the asynchronous multiple-access collision channel without feedback. The number of codewords in a strongly conflict-avoiding code is the number of potential users that can be supported in the system. In this paper, an improved upper bound on the size of strongly conflict-avoiding codes of length $$n$$n and weight three is obtained. This bound is further shown to be tight for some cases by direct constructions.

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.

Partially user-irrepressible sequence sets and conflict-avoiding codes

In this paper we give a partial shift version of user-irrepressible sequence sets and conflict-avoiding codes. By means of disjoint difference sets, we obtain an infinite number of such user-irrepressible sequence sets whose lengths are shorter than known results in general. Subsequently, the newly defined partially conflict-avoiding codes are discussed.

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.