A new upper bound and optimal constructions of equi-difference conflict-avoiding codes on constant weight
aa r X i v : . [ c s . I T ] F e b Noname manuscript No. (will be inserted by the editor)
A new upper bound and optimal constructions ofequi-difference conflict-avoiding codes on constant weight
Chun-e Zhao · Wenping Ma · Tongjiang Yan · Yuhua Sun
Received: date / Accepted: date
Abstract
Conflict-avoiding codes (CACs) have been used in multiple-access colli-sion channel without feedback. The size of a CAC is the number of potential usersthat can be supported in the system. A code with maximum size is called optimal. Theuse of an optimal CAC enables the largest possible number of asynchronous users totransmit information efficiently and reliably. In this paper, a new upper bound on themaximum size of arbitrary equi-difference CAC is presented. Furthermore, three op-timal constructions of equi-difference CACs are also given. One is a generalized con-struction for prime length L = p and the other two are for two-prime length L = pq . Keywords conflict-avoiding codes · equi-difference codes · optimal construction · exceptional code · non exceptional code Nowadays communication has become an indispensable part of people’s daily life.Coding plays an important role in kinds of communication systems, especially inmulti-access communication system. Multi-access channels are widely used in thefields of mobile and satellite communication networks. TDMA (time-division multi-ple address) is an important multi-access technique.In a TDMA system, the satellite working time is divided into periodic frames andeach frame is then divided into some time slots. In order to support user-irrepressibility,each user is assigned a protocol sequence which is derived by a CAC codeword. Soconflict-avoiding codes have been studied as protocol sequences for a multiple-access
This work is financially supported by the National Natural Science Foundation of China (No. 61902429),Fundamental Research Funds for the Central Universities (No. 19CX02058A), Shandong Provincial Nat-ural Science Foundation of China (ZR2019MF070). ∗ The corresponding author: [email protected] MaState Key laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China Chun-e Zhao et al. channel (collision channel) without feedback[1,2,3,4,5,6,7]. And the technical de-scription of such a multiple-access channel model can be found in [4,8]. The protocolsequence is a binary sequence and the number of ones in it is called its Hammingweight. The Hamming weight k of the sequence is the minimum weight requirementfor user-irrepressibility and it also means the maximal number of active users whocan send packets in the same time slot.So there are two different but complementary design goals in the literatures ofuser-irrepressible and conflict-avoiding sequences. The first one is minimizing thelength of the binary sequences for fixed potential users’ number N [9,10]. The secondone is maximizing the total number of potential users for fixed sequence length L andthe number of active users w . We concentrate on the second one in this paper.For fixed length L , many works are devoted to determine the maximal numberof potential users for Hamming weight three in [2,11,12,13,14,15]. Some optimalconstructions for Hamming weight four and five are presented in [16]. An asymptoticversion of this general upper bound can be found in [17]. A general upper bound onthe number of potential users for all Hamming weights is provided in [18].In [17], the asymptotic bound on the size of constant-weight conflict-avoidingcodes have been discussed. This is about the arbitrarily CACs, so the bound is notthe best for special cases. In this paper, we will focus on the equi-difference CACs.First, an upper bound on the maximum size of equi-difference CACs for constantweight is presented. This upper bound is lower than the former ones and its conciseexpression will greatly reduce the time complexity of validation. Secondly, three newconstructions of optimal equi-difference CACs are presented. Correspondingly, therange of CACs constructed will be enlarged. Third, the results show that these newcodewords’ size can reach this new upper bound. As a result, these CACs constructedare optimal and this new upper bound can be reached. Let P ( L , w ) be the set of all w -subsets of Z L = { , , ..., L − } and Z ∗ L = Z L \ { } .Given a w -subset I ∈ P ( L , w ) , we define the set of difference of I by d ∗ ( I ) = { j − i | i , j ∈ I , i = j } , where the j − i is modulo L . A conflict-avoiding code (CAC) C of length L and weight w is a subset C ⊂ P ( L , w ) satisfying the following condition d ∗ ( I j ) T d ∗ ( I k ) = /0 for any I j , I k ∈ C , j = k .Each element I ∈ C is called a codeword of length L and weight w .A codeword I is called equi-difference if the elements in I form an arithmeticprogression in Z L , i.e. I = { , g , g , ..., ( w − ) g } for some g ∈ Z L , where the product jg is reduced mod L , for j = , , , , ..., w −
1. The element g is called a generator ofthis codeword. For an equi-difference codeword I generated by g , the set of differenceis d ∗ ( I ) = {± g , ± g , ..., ± ( w − ) g } . utocorrelation distribution and 2-adic complexity 3 We note that the elements ± g , ± g , ..., ± ( w − ) g may not be distinct mod L . Hencein general we have | d ∗ ( I ) | ≤ w −
2. We adopt the terminology in [16] and say thata codeword I of weight w is exceptional if | d ∗ ( I ) | < w −
2. Let d ( I ) = d ∗ ( I ) ∪ { } ,then | d ∗ ( I ) | < w − | d ( I ) | < w − C are equi-difference, then C is called equi-difference.Let CAC e ( L , w ) denote the class of all equi-difference CACs of length L and weight w . For every code C ∈ CAC e ( L , w ) , C = C ∪ C always holds, where C = { I ∈ C , I isexceptional } and C = { I ∈ C , I is non exceptional } . The maximal size of some code C ∈ CAC e ( L , w ) is denoted by M e ( L , w ) , i.e. M e ( L , w ) = max {| C || C ∈ CAC e ( L , w ) } . A code C ∈ CAC e ( L , w ) is called optimal if | C | = M e ( L , w ) . An optimal code C ∈ CAC e ( L , w ) is called tight if S I ∈ C d ∗ ( I ) = Z ∗ L . [17]: | d ( A ) | ≥ | A | for any subset A in G. Lemma 2 [17]: Let w ( n ) denote the number of distinct prime divisors of n. For n ≥ and w ≥ , we have M ( n , w ) ≤ n − w − + w ( n ) Lemma 3 [18]: For L ≥ w ≥ ,M ( L , w ) ≤ ⌊ L − + F ( L , w ) w − ⌋ , (2) where F ( L , w ) : = max S ∈ ℓ ( L , w ) ∑ x ∈ S ( x − − x ⌈ w / x ⌉ + w ) , ℓ ( L , w ) : = { S ⊆ S ( L , w ) : gcd ( i , j ) = , ∀ i , j ∈ S , i = j } , S ( L , w ) : = { x ∈ { , , · · · , w − } : x divides L, and x ⌈ w / x ⌉ − x ≤ w − } . The two upper bounds listed above have their own merits and drawbacks. Usinga similar method to the one used in Theorem 3.7[20], we give a new upper bound forequi-difference CACs. It is easier to be reached than the first one and more easily todeal with problems than the second one.
Theorem 1
Let Ω ( L , w ) = { p is a divisor of L and w ≤ p < w − } . For n ≥ andw ≥ , then M e ( L , w ) ≤ L − + ∑ p ∈ Ω ∗ ( L , w ) ( w − − p ) w − , (3) where Ω ∗ ( L , w ) = { p ∈ Ω ( L , w ) | p is prime or p satisfies if gcd ( p , p ′ ) = for p ′ ∈ Ω ( L , w ) , p ≤ p ′ always holds } . Chun-e Zhao et al.
Proof
Let C be an ( L , w ) -equi-difference CAC, in which there are E exceptionalcodewords. Suppose C = C ∪ C , where C = { I ∈ C | I is non exceptional } and C = { I ∈ C | I is exceptional } . Then | C | = E . For i = , , · · · , E , denote the ith exceptional codeword by I i and let | d ∗ ( I i ) | = f i . Then we have the following inequal-ities: ( w − ) | C | + E ∑ i = f i ≤ L − ( w − )( | C | + E ) ≤ L − + E ∑ i = ( w − − f i ) (5)In fact, for every exceptional codeword I i with generator g i , d ( I i ) is a subgroup withgenerator gcd ( L , g i ) and then | d ( I i ) | is a divisor of L . Let | d ( I i ) | = p i , then ( w − )( | C | + E ) ≤ L − + E ∑ i = ( w − − p i ) . (6)For every two exceptional codewords I , I , d ∗ ( I ) T d ∗ ( I ) = /0 if and only their gen-erators are relatively prime. This implies that gcd ( | d ( I ) | , | d ( I ) | ) =
1. So each ele-ment in { p , p , · · · , p E } satisfies(i) p i | L ; (ii) gcd ( p i , p j ) =
1, for i = j ; (iii) w ≤ p i < w − E ∑ i = ( w − − p i ) ≤ ∑ p ∈ Ω ∗ ( L , w ) ( w − − p ) , | C | = | C | + E , then eq.(6) turns to be | C | ≤ L − + ∑ p ∈ Ω ∗ ( L , w ) ( w − − p ) w − . (7)So we have M e ( L , w ) ≤ L − + ∑ p ∈ Ω ∗ ( L , w ) ( w − − p ) w − . (8)Using this bound, we can deal with the Theorems 5-8 in [17] and Corollary 7 in[18] for equi-difference condition easily. And we also get the following Corollary1immediately. Corollary 1
Let L be an integer factorized as a b c d l, where l is not divisible by , , or . Then we haveM e ( L , w ) ≤ ⌊ L + ⌋ , for w = ⌊ L + ⌋ , for w = ⌊ L + ⌋ , for w = ⌊ L + ⌋ , for w = . Following, we will give three optimal constructions for equi-difference CACs. Oneillustrate the superiority of this new bound. The other two show the enlarged range ofoptimal equi-difference CACs constructed. utocorrelation distribution and 2-adic complexity 5
We use the following notation in [16]. For a subgroup H of G with | G || H | = f , if eachcoset H j of H contains exactly one element in { i , i , · · · , i f } for j = , , · · · , f , then { i , i , · · · , i f } is said to form a system of distinct representatives( SDR for short) of { H , H , · · · , H f } . Let Z × L = { a ∈ Z L | gcd ( a , L ) = } . condition 1 There exists a subgroup H of Z × L such that − ∈ H , | H | = | Z × L | ( w − ) and { , , · · · , w − } forms a SDR of H ’s cosets. condition 2 There exists a subgroup H of Z × L such that − ∈ H , | H | = | Z × L | ( w − ) and {± , ± , · · · , ± ( w − ) } forms a SDR of H ’s cosets.4.1 Optimal construction on equi-difference CACs of length L = p Lemma 4 [16]: Let p = ( w − ) m + be a prime number and suppose that { , , · · · , w − } forms a SDR of { H w − j ( p ) : j = , · · · , w − } . Let α be a primitive element in thefinite field Z p and let g = α w − . Then the m codewords of weight w generated by , g , g , · · · , g m − form an equi-difference ( ( w − ) m + , w ) − CAC.
The p ’s satisfying SDR in Lemma 4 are rare. Following, we will give a general-ized construction in which the range of p ’s will be enlarged. Theorem 2
Let p = ( w − ) ms + be a prime number and H a subgroup of Z ∗ p withorder m ( w − ) . Suppose that { , , · · · , w − } forms a SDR of { N , N , · · · , N w − } ,where N is a subgroup of H with order m and N j s are N ’s cosets in H. Let α bea primitive element of Z p and let g i j = α i + s ( w − ) j for ≤ i ≤ s − , ≤ j ≤ m − .Then the sm codewords of weight w generated by g i j form an optimal equi-difference ( ( w − ) ms + , w ) − CAC.Proof
Because H is a subgroup of Z ∗ p with order 2 m ( w − ) , then there exists anprimitive element α of Z p such that H = ( α s ) . Then Z ∗ p = s − [ i = α i H . Let g = α s ( w − ) and N = ( g ) be the subgroup of H with order 2 m . For { , , · · · , w − } forms a SDR of the cosets of N , then H = w − [ j = jN . And for the the reason that the order of N is 2 m and g m = α s ( w − ) m = α p − = −
1, so N = {± , ± g , ± g , · · · , ± g m − } . Let A = {± , ± , · · · , ± ( w − ) } , then H = ∪ w − j = jN = ∪ m − t = g t A . So Z ∗ p = s − [ i = α i H = s − [ i = m − [ j = α i g j A . Chun-e Zhao et al.
Let Γ ( C ) = { α i g j , ≤ i ≤ s − , ≤ j ≤ m − } be the set of generators of C , then I ( i , j ) = { , α i g i , α i g i , · · · , ( w − ) α i g i } , for 0 ≤ i ≤ s − , ≤ j ≤ m −
1. And d ∗ ( I ( i , j ) ) ∩ d ∗ ( I ( k , s ) ) = /0for ( i , j ) = ( k , s ) . So C = { I ( i , j ) | ≤ i ≤ s − , ≤ j ≤ m − } forms an equi-differenceCAC. The size of C is | C | = sm = m ( w − ) s w − = p − w − . By Theorem 1 we can see that C is an optimal equi-difference CAC. And Z ∗ p = s − [ i = w − [ j = d ∗ ( I ( i , j ) ) . So C is an optimal and tight equi-difference CAC.For s = Example 1
Let p = p = ( w − ) ms +
1, let w = , m = , s = α = Z ∗ p . Let N = ( α ) and H = ( α ) be the subgroup generated by α and α , respectively. N is a subgroup of H . We can check that { , , } forms a SDR of the cosets N , N , N in H . The 153codewords generated by the generators form an optimal (919,4)-CAC C . The set ofthe generators is Γ ( C ) = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } .4.2 Optimal constructions on equi-difference CACs of length L = pq After we have constructed CACs for prime length L = p based on Theorem 2, wewill give a recursive construction of CACs for two prime length L = pq in order toenlarge the range of CACs further. Theorem 3
Let C be an optimal tight ( p , w ) -equi-difference CAC with m code-words and C an optimal tight ( p , w ) -equi-difference CAC with m codewords. Thenset C = { I ( k , i ) , J ( j , j ) | ≤ k ≤ m , ≤ i ≤ p − , ≤ j ≤ m } utocorrelation distribution and 2-adic complexity 7 forms an ( p p , w ) optimal equi-difference CAC with m p + m codewords, whereI ( k , i ) = ( , a + ip , a + ip , · · · , a w − + ( w − ) ip ) for I k = ( , a , a , · · · , a w − ) ) ∈ C , J ( j , j ) = ( , b p , b p , · · · , b w − p ) for J j = ( , b , b , · · · , b w − ) ∈ C .Proof (1) d ∗ ( I ( k , i ) ) ∩ d ∗ ( I ( k , i ) ) = /0 for ( k , i ) = ( k , i ) Let I ( k , i ) = ( , g k + ip , g k + ip , · · · , ( w − ) g k + ( w − ) ip ) in C for I k =( , g k , g k , · · · , ( w − ) g k ) ∈ C . Then d ∗ ( I ( k , i ) ) = {± ( g k + ip ) , ± ( g k + ip ) , · · · , ± (( w − ) g k + ( w − ) ip ) } . For ( k , i ) = ( k , i ) , if there exists some xg k + xi p = yg k + yi p for − ( w − ) ≤ x , y ≤ ( w − ) , then xg k − yg k = ( yi − xi ) p ( mod p p ) (9). So xg k = yg k ( mod p ) . So x = y , g k = g k for xg k = yg k ∈ Z p ∗ and xg k ∈ d ∗ ( I k ) , yg k ∈ d ∗ ( I k ) . So k = k . Then in eq.(9), we have yi = xi ( mod p p ) . So x = y and i = i for − ( w − ) ≤ x , y ≤ ( w − ) and 0 ≤ i , i ≤ p −
1. This contractswith ( k , i ) = ( k , i ) . So d ∗ ( I ( k , i ) ) ∩ d ∗ ( I ( k , i ) ) = /0 for ( k , i ) = ( k , i ) .(2) d ∗ ( J ( j , j ) ) ∩ d ∗ ( J ( j , j ) ) = /0 for j = j Let J ( j , j ) = ( , b j p , b j p , · · · , ( w − ) b j p ) for J j = ( , b j , b j , · · · , ( w − ) b j ) ∈ C . Then d ∗ ( J ( j , j ) ) = {± b p , ± b p , · · · , ± ( w − ) b p } For j = j , if there exists − ( w − ) ≤ x , y ≤ ( w − ) , such that xb j p = yb j p ( mod p p ) . We also get that xb j − yb j = ( mod p ) .So xb j = yb j ∈ d ∗ ( J ( j , j ) ) ∩ d ∗ ( J ( j , j ) . This contractswith d ∗ ( J ( j , j ) ) ∩ d ∗ ( J ( j , j ) = /0. So d ∗ ( J ( j , j ) ) ∩ d ∗ ( J ( j , j ) ) = /0 for j = j .(3) d ∗ ( I ( k , i ) ) ∩ d ∗ ( J ( j , j ) ) = /0 for any k , i , j If there exists some x ( g k + ip ) = y ( b j p ( mod p p ) , then xg k = ( mod p ) which contracts with xg k ∈ d ∗ ( I k ) . So d ∗ ( I ( k , i ) ) ∩ d ∗ ( J ( j , j ) ) = /0. By (1)(2)(3) we getthat C is an equi-difference CAC.(4) | C | = m p + m . On the other aspect, by Lemma 1 | C | ≤ ⌊ L − w − ⌋ = p p − w − = p ( p − )+ p − w − = p − w − p + p − w − = m p + m .So C is an optimal equi-difference CAC.Further more, we will give another optimal construction of CACs for two primelength L = pq . It indicates that the new upper bound given in Theorem 2 is lowerthan the known one listed in Lemma 2 and is more convenient than the one listed inLemma 3. Theorem 4
Let L = pq, where q = ( w − ) f + and w ≤ p ≤ ( w − ) are bothprimes. If L and q satisfy condition 1 or condition 2, then there exists an optimalequi-difference ( L , w ) − CAC C with | C | = p f + .Proof It is well known that Z × L is a multiplicative group. Let ( p ) = { kp ( mod L ) | k ∈ Z } and ( q ) = { kq ( mod L ) | k ∈ Z } be the additive subgroups of Z L . It is clear that Z ∗ L = Z × L ∪ ( p ) ∗ ∪ ( q ) ∗ , where ( p ) ∗ = ( p ) \ { } , ( q ) ∗ = ( q ) \ { } . Take L , q satisfying Chun-e Zhao et al. condition 1 for example. We consider the elements in Z × L , ( p ) ∗ and ( q ) ∗ , respectively.(1) Elements in Z × L .There exists a subgroup H of Z × L such that − ∈ H and { , , · · · , ( w − ) } formsa SDR of the cosets. Noted | H | by 2 s . Let α be the generator of H . Select each g i = α i ∈ H as the generator of I i = { , g i , g i , · · · , ( w − ) g i } , i = , , · · · , s , then d ∗ ( I i ) = {± g i , ± g i , · · · , ± ( w − ) g i } . If i = j , ≤ i , j ≤ s , then d ∗ ( I i ) ∩ d ∗ ( I j ) = φ and ∪ si = d ∗ ( I i ) = Z × L . (2) Elements in ( p ) ∗ .Because | ( p ) ∗ | = | Z ∗ q | and q satisfies condition 1, then there exists a subgroup N of Z ∗ q such that − ∈ N and { , , · · · , ( w − ) } forms a distinct representatives ofthe cosets of N . Noted | N | by 2 t . Let β be the generator of N . Select each b i = β i p ( mod L ) ∈ ( p ) ∗ as the generator to construct a codeword J i = { , b i , b i , · · · , ( w − ) b i } , i = , , · · · , t , then d ∗ ( J i ) = {± b i , ± b i , · · · , ± ( w − ) b i } . If i = j , ≤ i , j ≤ t ,then d ∗ ( J i ) ∩ d ∗ ( J j ) = φ and ∪ ti = d ∗ ( I i ) = Z ∗ q . (3) Elements in ( q ) ∗ .If the codeword generated by q is noted by K , then | d ∗ ( K ) | = | ( q ) ∗ | = p − ≤ ( w − ) − < ( w − ) so K is exceptional and ( q ) ∗ = d ∗ ( K ) . Then C = { I , I , · · · , I s , J , J , · · · , J t , K } forms an equi-difference conflict-avoidingcode. The size of C is | C | = s + t + = ( p − )( q − ) ( w − ) + q − ( w − ) + = p f + . On the other hand, by Theorem 1, the size of the code satisfies | C | ≤ L − + ∑ p ∈ Ω ∗ ( L , w ) ( w − − p ) w − = L − + ( w − − p ) w − = L − p w − + = p ( ( w − )) f w − + = p f + Example 2
Let L = , w = , p = , q = , f = H = ( ) is a subgroup satisfies condition 2, and s = , g =
45. The 34codewords generated by the generators form an optimal (671,11)-CAC C . The set ofthe generators is Γ ( C ) = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } . utocorrelation distribution and 2-adic complexity 9 In this paper, we first give a new upper bound of equi-difference CACs. Using thisbound, it is easier to be reached and is easier to deal with some problems. Secondly,we give three optimal constructions of equi-difference CACs. One shows the supe-riority of the new upper bound and the other two make the range of optimal CACsconstructed enlarged.