We show that when gcd(n,w) = 1, the set of binary words of length n and weight w can be partitioned to give n maximal w-weight codes. It follows that under the same hypothesis, the least cardinal of a maximal constant weight code is at most 1/n times n choose w

Berkesch, Christine

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Rose-Hulman Undergraduate Mathematics Journal Volume 5 Issue 1 Article 1 An Array of Disjoint Maximal Constant Weight Codes Christine Berkesch Butler University, cberkesc@butler.edu Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Berkesch, Christine (2004) "An Array of Disjoint Maximal Constant Weight Codes," Rose-Hulman Undergraduate Mathematics Journal: Vol. 5 : Iss. 1 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol5/iss1/1 An Array of Disjoint Maximal ConstantWeight CodesChristine BerkeschApril 21, 2004AbstractWe show that when gcd(n,w) = 1, the set of binary words of length n and weight w can be partitionedto give n maximal w-weight codes. It follows that under the same hypothesis, the least cardinal of amaximal constant weight code is at most 1n`nw´.1 IntroductionCommunication plays a vital role in the world today. With rapid advances intechnology, an increasing amount of information must be exchanged between theindividual and machines. When a message is sent through a channel, interference,called noise, can cause errors. The noise includes human error, crosstalk, lightning,thermal noise, or impulse noise. Whether this information is being saved to acomputer disk, sent via FAX, or transmitted in radio or mobile communication,the presence of inaccuracies can have massive consequences. Fortunately, theseerrors can be overcome when an initial language, or code, is constructed withcertain error-detecting or correcting features. Coding theory is a relatively newbranch of mathematics devoted to discovering languages, called codes, which arebeing successfully employed to transmit messages accurately and in a most efficientmanner.Coding theory began in 1948 after Claude Shannon, an employee at Bell Lab-oratories in the USA, showed that it was possible to encode messages so that thenumber of extra bits was as small as possible. This resulted in the publicationof the classic paper The Mathematical Theory of Communication in the Bell Sys-tem Technical Journal. Two years later, Richard Hamming, another employee atBell Labs, devised a family of single error-correcting codes in order to enhance theperformance of a computer [1].A popular example which demonstrates the value of coding theory is the com-munication of Mariner 9 with NASA. This was a probe sent to Mars in 1971. In1order to transmit gray-scale pictures of that planet, individual pixels were sent ona scale from 0 to 63. To send these numbers, NASA implemented a Reed-Mullercode capable of correcting up to 7 errors in a single 32-bit word (consisting of 6data bits and 26 check bits). Mariner 9 was able to transmit over 16,000 bits persecond back to Earth.An everyday example of error-correcting codes can be found in the compactdisc. On CDs, the signal is encoded digitally. To prevent skips from scratches,two “interleaved” codes are used which can correct up to 4,000 consecutive errors(about 2.5 mm of track) [4].2 DefinitionsLet F2 denote the finite field of order 2, and Fn2 the n-dimensional vector space overF2. The elements of Fn2 are called binary words of length n. The weight of a wordx = (x1, x2, . . . , xn), denoted ||x||, is defined to be |{i : xi = 0}|. The Hammingdistance H between words x and y ∈ Fn2 is the number of positions in which theydiffer. In other wordsH(x,y) := |{i : 1 ≤ i ≤ n, xi = yi}| = ||x− y||.A non-empty subset C of Fn2 is called a binary code, and the elements of Care binary codewords. C is called an (n, d) binary code if d ≤ H(x,y) for allx,y ∈ C,x = y. The minimum distance in a code C is defined to be min{H(x,y) :x,y ∈ C,x = y}.The codewords of length n in a binary code are sometimes referred to as n-bitcodewords. When dealing with such codes, 0 and 1 denote the n-bit words of all0’s and 1’s, respectively.Let P be a property of binary codes. A code C is said to be maximal P providedC has P and no code having P properly contains C. A code C is optimal P providedC has property P and for any code D having P , |D| ≤ |C|.LetX denote the set {1, 2, 3, . . . , n}, and P (X) the power set ofX. The functionr : Fn2 → P (X), defined by r(x) = {i : xi = 1} is a one-to-one correspondencebetween Fn2 and P (X). Thus, each binary word of length n will be identified witha subset of X. Let(Xw)denote the set of subsets associated with binary words ofweight w.For two sets A and B, their symmetric difference is AB := (A−B)∪(B−A).Observe that H(x,y) = |r(x) r(y)|.23 Constant Weight Binary CodesAn (n, d) code C is said to have constant weight w, and is called an (n, d, w) code,if every word in C has weight w. A(n, d) and A(n, d, w) respectively denote thecardinality of an optimal (n, d) and an optimal (n, d, w) code. Similarly, a(n, d) anda(n, d, w) respectively denote the least possible cardinality of a maximal (n, d) codeand a maximal (n, d, w) code. In general, the values of all four of these functionsare unknown.In an exhaustive study of the lower bounds for A(n, d, w), A. E. Brouwer, et al.[2] found an elegant method to partition the set of binary words of length n andweight w into n (n, 4, w) codes, thus proving that A(n, 4, w) ≥ 1n(nw). The purposeof this paper is to show that if gcd(n,w) = 1, then each of the n codes defined in[2] is, in fact, maximal (n, 4, w). Before proving this result, we first briefly sketchthe construction given in [2].Theorem 1. A(n, 4, w) ≥ 1n(nw)Proof. Let Zn be the group of integers modulo n, and consider the functionf :(Xw) → Zn, defined by f(A) = (∑a∈A a) (mod n). For an i, 0 ≤ i ≤ n − 1,let Ci = f−1(i). Let A,B ∈ Ci, A = B. We will show that |A B| ≥ 4. Clearly|AB| = 2w − 2|A ∩B|, a positive even integer. Now if |AB| = 2, then thereexist a0 ∈ A and b0 ∈ B such that A B = {a0, b0}. Since∑a∈A a =∑b∈B b = i(mod n), we have that a0 = b0 (mod n). But that is impossible because a0 = b0and both a0 and b0 belong to the set X = {1, 2, . . . , n}. It follows that |AB| ≥ 4;thus, each Ci is an (n, 4, w) code. As the collection {C0, C1, . . . , Cn−1} is obviouslya partition of(Xw)and |(Xw)| = (nw), therefore some Ci must have at least1n(nw)elements. Theorem 2. If 2 ≤ w < n and gcd(n,w) = 1, then (Xw)can be partitioned into nmaximal (n, 4, w) codes.Proof. We will prove that under the given hypothesis, each of the sets Ci, 0 ≤i ≤ n − 1, defined in the proof of Theorem 1 is a maximal (n, 4, w) code. Wedo this by showing that for each i and for any A ∈ (Xw), if A /∈ Ci, then there isa B ∈ Ci such that H(A,B) = 2. Fix an i and an A, as stated. Determine j,0 ≤ j ≤ n − 1, such that A ∈ Cj and then let k = (i − j)(mod n). As A /∈ Ci,j = i, and thus 0 < k < n. For each a ∈ A, let a∗ = (a + k)(mod n). If for somea0 ∈ A, a∗0 /∈ A, then by deleting a0 from A and inserting a∗0 in its place, we obtain3a set B as desired. To see why, note that A B = {a0, a∗0}, and so |A B| = 2.Also, summing modulo n, we have∑b∈B b =∑a ∈ A+a∗0−a0 = j+k = i. Thus,B ∈ Ci.To complete the proof, we show that under the given hypothesis, there existsan a ∈ A for which a∗ /∈ A. By way of contradiction, suppose such an a does notexist. Let 〈k〉 be the subgroup of Zn generated by k. Then for each a ∈ A, thecoset 〈k〉 + a is contained in A. Therefore, A is a union of a set of cosets of 〈k〉in Zn. As the order of the subgroup 〈k〉 is ng where g = gcd(k, n), it follows thatthe cardinality of A, namely w, is a multiple of ng. This, in view of the hypothesisthat gcd(n,w) = 1 implies that ng= 1, so n = gcd(n, k). But then k ≥ n, acontradiction. Corollary. If gcd(n,w) = 1, then a(n, d, w) ≤ 1n(nw).The proof of Theorem 2 implies that if gcd(n,w) = g > 1, then at least one ofthe Ci is not a maximal (n, 4, w) code. To see this, let w = gw1 and n = gn1, andnote that the subgroup 〈n1〉 of Zn is of order g. Choose any w1 of the cosets of 〈n1〉in Zn, and let A be their union, so that A ∈(Xw). Then, for each a ∈ A, (a + n1)(mod n) ∈ A. Thus, if A ∈ Cj and if i = (j+n1) (mod n), then A /∈ Ci even thoughA is at a distance at least 4 from each member of Ci, as shown in the proof of The-orem 1. Thus, Ci is not a maximal (n, 4, w) code, and we have shown the following.Theorem 3. If gcd(n,w) > 1, then at least one Ci is not a maximal (n, 4, w) code.An interesting interpretation of Theorem 2 in terms of “graph-coloring” is asfollows: If gcd(n,w) = 1, it is possible to color each binary word of length n andweight w in one of n given colors so that each of the monochromatic classes is amaximal (n, 4, w) code. It is noteworthy that similar “color” partitions into optimalcodes do not always exist. This is due to the fact that A(n, d) and A(n, d, w)generally fail to divide 2n and(nw), respectively. For example, A(9, 4) = 40 whichdoes not divide 29, so a monochromatic partition of F92 into optimal (9, 4) codesdoes not exist. Similarly, A(11, 4, 4) = 35, which does not divide(114), so the setof binary words with length 11 and weight 4 cannot be partitioned into optimal(11, 4, 4) codes.Acknowledgements. This research was done during Summer 2003 at ButlerUniversity under the guidance of Dr. Prem Sharma as a part of the Butler SummerInstitute, funded by a Holcomb Undergraduate Grant.4References[1] A History of Fundamental Mathematics Research at Bell Labs (1998).(http://cm.bell-labs.com/cm/ms/departments/fm/history.html)[2] A. E. Brouwer, James B. Shearer, N. J. A. Sloane, and Warren D. Smith,A New Table of Constant Weight Codes, IEEE Transactions on InformationTheory, Vol. 36, No. 6, November 1990.[3] J. H. van Lint, Introduction to Coding Theory, 3rd Edition, Springer-Verlag,Berlin 1999.[4] Richard Pinch, Coding theory: the first 50 years, PASS Maths, Issue No. 3:September 1997. (http://plus.maths.org/issue3/codes/index.html)[5] E. M. Rains and N. J. A. Sloane, Table of Constant Weight Binary Codes.(http://www.research.att.com/ njas/codes/Andw/)5

An Array of Disjoint Maximal Constant Weight Codes

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An Array of Disjoint Maximal Constant Weight Codes

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