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BCH codes are good for distance multicolorings with radio frequencies in hypercubes asymptotically
BCH codes are good for distance multicolorings with radio frequencies in hypercubes asymptotically Zdzislaw Skupie n
Faculty of Applied Mathematics, University of Mining and Metallurgy AGH al. Mickiewicza 30, 30{059 Krakow, Poland e-mail: [email protected] March 26, 2001
Abstract
It is announced that binary primitive narrow-sense BCH codes or their shortenings have size asymptotically within a constant factor below the largest possible size among binary codes of the same length, n, and the same minimum distance, d, as n ! 1 while d is constant. Consequently the asymptotic growth of distance multicoloring parameters in hypercubes is determined. Connections with radio frequencies allocation problem are discussed.
1 The maximum error-correcting codes BCH codes are important error-correcting codes which are linear and cyclic, named after Bose and Chaudhuri (1960) and Hocquenghem (1959), the discoverers of the binary BCH codes. The aim of this note is to show that binary BCH codes are nearly optimal in an obscure case when their minimum distance d is xed and the length n ! 1, which makes the long codes useless for detection of transmission errors. However, this appears su cient for determining the exact order of growth of parameters for vertex and edge distance 1
colorings in growing hypercubes. Applications to distance multicolorings which can model the frequency assignment in cellular telecommunication are dealt with. Results on coding we refer to in what follows can be found in the monograph 2] by MacWilliams and Sloane, for distance colorings we refer to author's paper 4]. Recall the following notation. Let A(n d) be the maximum size (= the number of codewords) in any binary code (possibly nonlinear) of length n and minimum distance d. Let B (n d) be the corresponding maximum size among linear codes with the same parameters n and d. Moreover, the name an (n M d) code stands for a binary code of length n, size (or cardinality) M , and minimum distance d. A linear (n M d) code C is named an n k d] code where k = log2 M is an integer called the dimension of C . Finding A(n d) (or rather a code which realizes A(n d)) is clearly among the most basic problems. Advantages of linearity motivate the study of B (n d). A method for determining the maximum linear code and its size B (n d) for d n 2d is presented by Venturini. Just in this case A(n d) is determined by the Plotkin bound and its relative sharpness shown by Levenshtein. Nevertheless, even in this case it is an open problem to nd a nontrivial lower bound for B (n d) in terms of A(n d). We are going to contribute to this problem. Let (2n)!! stand for 2n n!. Recall that the oor bxc of a real number x is the integer part of x. Moreover, the ceiling dxe = bxc.
Theorem 1
B (n d)
as n ! 1 and d is constant.
> (2b(dA(n1)d=)2c)!!
Even though \long BCH codes are bad"|in the sense that BCH codes are unable to keep both ratios k=n and d=n away from zero as n is large, cp.
3]|they are large enough as is shown below. Theorem 2 Let C~ be a primitive narrow-sense BCH code with designed odd distance d. If either C = C~ or C is got by shortening C~ by less than a half to the length n d, then the size jCj of C exceeds a constant part of the largest possible size among binary codes of the same length n and the minimum distance just d, as n ! 1 and d is kept constant. In fact, if 2m1 n 2
2m 1 so that 2m 1 is the length and d = 2 + 1 is the designed distance of the BCH code C~ in question and n d then n B (n 2 + 1) jCj 2nm (22n) for all n > A(n22 !+ 1) as n ! 1: (1)
The proof is based on classical properties like the Hamming bound, BCH bound and theorems by Farr and Mann on true minimum distance and exact size of BCH codes.
2 Distance multicolorings In what follows by a graph we mean a loopless multigraph. To color elements like vertices (or edges) of a graph means to assign a color to each element. A color class of a coloring is the set of unicolored elements. In proper (classical) coloring unicolored elements are at distance larger than one. The distance between two edges in a graph G is de ned to be their vertex distance in the line graph L(G). If no two unicolored elements are at distance d or less, the coloring is called d+ -distance coloring, d 2 N . Then a color class is called either a d+ -matching if edges are colored or a d+ -independent set if vertices are colored. A d+ -matching number, d (G), and d+ -independence number, d(G), are the largest possible sizes of the corresponding color classes among colorings of G. Given a function f from elements to nonnegative integers, a d+ -distance f -coloring of G consists in assigning f (x) colors to each element x so that for any two elements at distance d or less all assigned colors be distinct. The problem is to determine the corresponding (d f )-chromatic parameters: the chromatic number fd (G) and chromatic index qdf (G). These are d+ -distance chromatic parameters d(G) and qd(G) if all values of f are 1. Finding the (d f )-chromatic number fd (G) can model optimal stationary assignment of radio frequencies (colors) from a limited band to the regions (cells = vertices) x where edges of G join neighboring cells (at unit distance apart) and d is the upper bound on the distance within which interferences occur. A similar problem of so-called call chromatic number, denoted by CRf (G), with the constraint that all colors within any ball B (x R) of radius 3
R be di erent is quoted by Baldi 1]. It can be seen that the function fd f suits better than CR to applications in cellular telecommunication. Let Qt stand for a graph of t-dimensional cube. Recall that in asymptotic
notation denotes the exact order of growth. In what follows we replace f by an integer K which is assumed to be the only value of the function f . Then multicoloring is equivalent to classical coloring with disjoint K -sets as single colors. Theorem 3 If the multiplicity of colors K and distance bound d are xed natural numbers, and if the dimension t ! 1 then Kd (Qt) = K (tbd=2c) qdK (Qt) = K (tb(d+1)=2c): The proof reduces to K = 1 and then crucial are the inequalities 2t 2t t2t1 q (Q ) t2t1 ( Q ) A(t d + 1) d t B (t d + 1) A(t 1 d) d t B (t 1 d) established in Skupie n 4] together with d(Qt) = A(t d + 1) and d(Qt) = A(t 1 d). Upper bounds therein come from colorings whose color classes are translates of linear codes with corresponding lengths and minimumdistances. On the other hand, as is noted in 4], using some translates of the (nonlinear) punctured Preparata code P (2r) (with r > 1) which is an (n 2n+14r 5) code whose size equals A(n 5) with n = 22r 1 can give equalities 4(Qn) = (n + 1)2=2 and q5(Qn+1 ) = (n + 1)3=2 for n = 4r 1 15.
References
1] P. Baldi, On a generalized family of colorings, Graphs Comb., 6 (1990) 95{110.
2] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam et al., 1981.
3] S. Lin and E.J. Weldon, Jr., Long BCH codes are bad, Info. and Control, 11 (1967) 445{451.
4] Z. Skupie n, Some maximum multigraphs and edge/vertex distance colourings, Discuss. Math.{Graph Theory 15 (1995) 89{106. 4
Faculty of Applied Mathematics, University of Mining and Metallurgy AGH al. Mickiewicza 30, 30{059 Krakow, Poland e-mail: [email protected] March 26, 2001
Abstract
It is announced that binary primitive narrow-sense BCH codes or their shortenings have size asymptotically within a constant factor below the largest possible size among binary codes of the same length, n, and the same minimum distance, d, as n ! 1 while d is constant. Consequently the asymptotic growth of distance multicoloring parameters in hypercubes is determined. Connections with radio frequencies allocation problem are discussed.
1 The maximum error-correcting codes BCH codes are important error-correcting codes which are linear and cyclic, named after Bose and Chaudhuri (1960) and Hocquenghem (1959), the discoverers of the binary BCH codes. The aim of this note is to show that binary BCH codes are nearly optimal in an obscure case when their minimum distance d is xed and the length n ! 1, which makes the long codes useless for detection of transmission errors. However, this appears su cient for determining the exact order of growth of parameters for vertex and edge distance 1
colorings in growing hypercubes. Applications to distance multicolorings which can model the frequency assignment in cellular telecommunication are dealt with. Results on coding we refer to in what follows can be found in the monograph 2] by MacWilliams and Sloane, for distance colorings we refer to author's paper 4]. Recall the following notation. Let A(n d) be the maximum size (= the number of codewords) in any binary code (possibly nonlinear) of length n and minimum distance d. Let B (n d) be the corresponding maximum size among linear codes with the same parameters n and d. Moreover, the name an (n M d) code stands for a binary code of length n, size (or cardinality) M , and minimum distance d. A linear (n M d) code C is named an n k d] code where k = log2 M is an integer called the dimension of C . Finding A(n d) (or rather a code which realizes A(n d)) is clearly among the most basic problems. Advantages of linearity motivate the study of B (n d). A method for determining the maximum linear code and its size B (n d) for d n 2d is presented by Venturini. Just in this case A(n d) is determined by the Plotkin bound and its relative sharpness shown by Levenshtein. Nevertheless, even in this case it is an open problem to nd a nontrivial lower bound for B (n d) in terms of A(n d). We are going to contribute to this problem. Let (2n)!! stand for 2n n!. Recall that the oor bxc of a real number x is the integer part of x. Moreover, the ceiling dxe = bxc.
Theorem 1
B (n d)
as n ! 1 and d is constant.
> (2b(dA(n1)d=)2c)!!
Even though \long BCH codes are bad"|in the sense that BCH codes are unable to keep both ratios k=n and d=n away from zero as n is large, cp.
3]|they are large enough as is shown below. Theorem 2 Let C~ be a primitive narrow-sense BCH code with designed odd distance d. If either C = C~ or C is got by shortening C~ by less than a half to the length n d, then the size jCj of C exceeds a constant part of the largest possible size among binary codes of the same length n and the minimum distance just d, as n ! 1 and d is kept constant. In fact, if 2m1 n 2
2m 1 so that 2m 1 is the length and d = 2 + 1 is the designed distance of the BCH code C~ in question and n d then n B (n 2 + 1) jCj 2nm (22n) for all n > A(n22 !+ 1) as n ! 1: (1)
The proof is based on classical properties like the Hamming bound, BCH bound and theorems by Farr and Mann on true minimum distance and exact size of BCH codes.
2 Distance multicolorings In what follows by a graph we mean a loopless multigraph. To color elements like vertices (or edges) of a graph means to assign a color to each element. A color class of a coloring is the set of unicolored elements. In proper (classical) coloring unicolored elements are at distance larger than one. The distance between two edges in a graph G is de ned to be their vertex distance in the line graph L(G). If no two unicolored elements are at distance d or less, the coloring is called d+ -distance coloring, d 2 N . Then a color class is called either a d+ -matching if edges are colored or a d+ -independent set if vertices are colored. A d+ -matching number, d (G), and d+ -independence number, d(G), are the largest possible sizes of the corresponding color classes among colorings of G. Given a function f from elements to nonnegative integers, a d+ -distance f -coloring of G consists in assigning f (x) colors to each element x so that for any two elements at distance d or less all assigned colors be distinct. The problem is to determine the corresponding (d f )-chromatic parameters: the chromatic number fd (G) and chromatic index qdf (G). These are d+ -distance chromatic parameters d(G) and qd(G) if all values of f are 1. Finding the (d f )-chromatic number fd (G) can model optimal stationary assignment of radio frequencies (colors) from a limited band to the regions (cells = vertices) x where edges of G join neighboring cells (at unit distance apart) and d is the upper bound on the distance within which interferences occur. A similar problem of so-called call chromatic number, denoted by CRf (G), with the constraint that all colors within any ball B (x R) of radius 3
R be di erent is quoted by Baldi 1]. It can be seen that the function fd f suits better than CR to applications in cellular telecommunication. Let Qt stand for a graph of t-dimensional cube. Recall that in asymptotic
notation denotes the exact order of growth. In what follows we replace f by an integer K which is assumed to be the only value of the function f . Then multicoloring is equivalent to classical coloring with disjoint K -sets as single colors. Theorem 3 If the multiplicity of colors K and distance bound d are xed natural numbers, and if the dimension t ! 1 then Kd (Qt) = K (tbd=2c) qdK (Qt) = K (tb(d+1)=2c): The proof reduces to K = 1 and then crucial are the inequalities 2t 2t t2t1 q (Q ) t2t1 ( Q ) A(t d + 1) d t B (t d + 1) A(t 1 d) d t B (t 1 d) established in Skupie n 4] together with d(Qt) = A(t d + 1) and d(Qt) = A(t 1 d). Upper bounds therein come from colorings whose color classes are translates of linear codes with corresponding lengths and minimumdistances. On the other hand, as is noted in 4], using some translates of the (nonlinear) punctured Preparata code P (2r) (with r > 1) which is an (n 2n+14r 5) code whose size equals A(n 5) with n = 22r 1 can give equalities 4(Qn) = (n + 1)2=2 and q5(Qn+1 ) = (n + 1)3=2 for n = 4r 1 15.
References
1] P. Baldi, On a generalized family of colorings, Graphs Comb., 6 (1990) 95{110.
2] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam et al., 1981.
3] S. Lin and E.J. Weldon, Jr., Long BCH codes are bad, Info. and Control, 11 (1967) 445{451.
4] Z. Skupie n, Some maximum multigraphs and edge/vertex distance colourings, Discuss. Math.{Graph Theory 15 (1995) 89{106. 4