Onthe reconstructionof perfect codes

Transcription

1 Discrete Mathematics 256 (2002) Note Onthe reconstructionof perfect codes Olof Heden Department of Mathematics, KTH, S-00 44, Stockholm, Sweden Received 3 April 200; received inrevised form 5 October 200; accepted 29 October 200 Abstract We show how to reconstruct a perfect -error correcting binary code of length n from the code words of weight (n + )=2. c 2002 Elsevier Science B.V. All rights reserved.. Introduction We may dene perfect -error correcting binary codes in the following way: Consider a direct product Z = Z 2 Z 2 Z 2 of the rings Z 2 = 0; }. The elements of this direct product will be called words of length n. The weight of a word c, w(c), will be the number of non-zero components of c. The distance betweentwo words c and c, d(c; c ), will be the weight of the word c c. A perfect -error correcting binary code of length n is a subset C of Z2 n satisfying the following condition: To any v Z n 2 there is an unique c C with d(c; v) 6 : (Trivial counting arguments gives that the only possible value for the length of a perfect -error correcting binary code is n =2 m.) There are many dierent constructions of perfect -error correcting binary code and it seems to be hard to enumerate and classify them all, see [6]. For a perfect -error correcting binary code C, let C(k) denote the set of words of C of weight k. Avgustinovich [] has showed that if C((n +)=2) = C ((n +)=2) for two binary perfect -error correcting codes C and C of length n then C = C. His proof does not show how to reconstruct a perfect -error correcting binary code of length n from the words of the weight (n +)=2. The purpose of this note is to solve that problem. address: olohed@math.kth.se (O. Heden) X/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S X(02)0039-6

2 480 O. Heden / Discrete Mathematics 256 (2002) Preparations Below, a perfect code always will be a perfect -error correcting binary code in Z n 2. We consider a group algebra R[x ;x 2 ;:::;x n ]. The elements of this group algebra are polynomials r(x ;x 2 ;:::;x n )= v Z n 2 r v x v xv2 2 :::xvn n ; v=(v ;v 2 ;:::;v n ); () where the elements r v, for v Z, belongs to the set of real numbers R. The multiplicationof polynomials inthis group algebra is givenby the multiplication of monomials and extended to multiplication of polynomials in the usual way. If (w ;w 2 ;:::;w n ) is the sum of (u ;u 2 ;:::;u n )and(v ;v 2 ;:::;v n )inz2 n then x v xv2 2 :::xvn n x u xu2 2 :::xun n = x w xw2 2 :::xwn n : When only addition is concerned we may also consider R[x ;x 2 ;:::;x n ] as a vector space over the real numbers. This vector space is generated by the monomials x v xv2 2 :::xvn n,(v ;v 2 ;:::;v n ) Z. The dimension of this vector space thus equals 2n. Let y t (x ;x 2 ;:::;x n ), for t Z, denote the polynomial y t (x ;x 2 ;:::;x n )= ( x i ) ti ( + x i ) ti ; t=(t ;t 2 ;:::;t n ): Lemma. The polynomials y t (x ;x 2 ;:::;x n ), for t Z, constitute a base of the vector space R[x ;x 2 ;:::;x n ]. Proof. There are polynomials y t (x ;:::;x n ). It thus suces to show that any polynomial r(x ;x 2 ;:::;x n )ofr[x ;x 2 ;:::;x n ] has an unique expansion r(x ;x 2 ;:::;x n )= A t y t (x ;:::;x n ); (2) t Z2 n where A t R for t Z. If we inthe above equality substitute x i = if di =0; if d i =; d=(d ;d 2 ;:::;d n ) Z n 2 ; (3) thenwe get from the Eq. () that where A d = r v ( ) v d ; (4) v Z2 n (v ;v 2 ;:::;v n )(d ;d 2 ;:::;d n )=v d + v 2 d v n d n : Thus there is only one possibility for the coecient A d and the lemma is proved.

3 In[2] we proved that y t (x ;:::;x n )y t (x ;:::;x n )= O. Heden / Discrete Mathematics 256 (2002) yt (x ;:::;x n ) if t = t ; 0 if t t : Hence, the coecients A t, t Z, inexpansion(2) will be called the fourier coecients of the polynomial r(x ;:::;x n ). To a subset C of Z2 n we associate the polynomial C(x ;x 2 ;:::;x n )= c C x c xc2 2 :::xcn n ; c=(c ;c 2 ;:::;c n ): We will say that the fourier coecients of the polynomial C(x ;x 2 ;:::;x n ) are the fourier coecients of the set C. Let e i denote the word with support i. (The support of a word is the set of non-zero coordinates of the word.) For any subset C of Z n 2, we denote by C(i) the set C (i) = c + e i c C}: We say that the set C (i) is obtained from the set C by switching the ith coordinate. Lemma 2. If the set C has the fourier coecients A t, t Z, and C(i) the fourier coecients A (i) t, t Z, then, for i =; 2;:::;n, A (i) t = At if t i =0; A t if t i 0: t =(t ;t 2 ;:::;t n ) Z n 2 : Proof. Switching the ith coordinate in Z n 2 corresponds in the group algebra R[x ;x 2 ;:::; x n ] to multiplicationwith the monomial x i.asx i x i =, we get that x i ( x i )= ( x i ) an d x i ( + x i )=(+x i ) and the lemma follows. Let D denote the set of words of Z of weight (n +)=2. The following result was proved by Roos [4] with slightly dierent methods. Theorem. If C is a perfect -error correcting binary code of length n then there are integers A 0 and A d, d D, such that C(x ;:::;x n )= A 0 ( + x i )+ d D A d ( + x i ) di ( x i ) di : (This result was generalized in [2] to perfect codes over anarbitrary alphabet.) Proof. Assume that C has the fourier coecients A 0 t, t Z. Denote by A(i) t, t Z2 n and i =; 2;:::;n, the fourier coecient of the switched code C (i).

4 482 O. Heden / Discrete Mathematics 256 (2002) We note that if B t, t Z, are the fourier coecients of the set Zn 2, then B 0 = and B t = 0 for t Z\0}. As the union of the code C with the codes C (i), i =; 2;:::;n, equals Z2 n and as these n + codes are disjoint, we thus get that, for any t Z\0}, n i=0 A (i) t =0: The theorem now follows from Lemma 2. If we let x i = for i =; 2;:::;n in(2) and(3) we will get that C = c C =C(; ;:::;) = A 0 : Let d denote the set of words that are orthogonal to the word d=(d ;d 2 ;:::;d n ) in Z n 2, i.e. d =(v ;v 2 ;:::;v n ) d v + d 2 v d n v n 0 (mod 2)}: We get from Eq. (4) that A d =2 d C C : (5) Hergert [3] observed that if d 0 is orthogonal to all words of C, then w(d)= (n +)=2. Hence, if C denotes the linear span of the words of C, thenwe may conclude from (5) that d C if and only if A d = C : (6) Let S (v) denote the set of words at distance at most one from the word v. This subset of Z isa-sphere around the word v. Weighted perfect codes will be essential in the proof of our result. A weighted perfect code f may be considered as a function from Z to the set of real numbers R with the property f(v) = v S (a) for all words a Z. The function f that describes anordinary perfect code C will be the function if v C; f(v)= 0 if v= C: To the weighted perfect code f we associate the polynomial f(x ;x 2 ;:::;x n )= v Z n 2 f(v)x v xv2 2 :::xvn n v =(v ;v 2 ;:::;v n ):

5 O. Heden / Discrete Mathematics 256 (2002) With the same arguments as in the proof of Theorem we get Proposition. A function f from Z n 2 to R is a weighted perfect code if and only if f(x ;:::;x n )= A 0 ( + x i )+ d D A d ( + x i ) di ( x i ) di for some unique real numbers A d, d D, and with A 0 = = S (0). A word p is a period of a weighted perfect code f if f(v + p)=f(v) for all words v of Z. The set of periods of a weighted perfect code f is the kernel of f, ker(f). Trivially, the kernel of a weighted perfect code is a subspace of Z. Let p be any xed word and let f be a weighted perfect code. Let g be the weighted perfect code dened by g(v+p)=f(v) for all v Z. We denote this code g by f+p. The following lemma is an immediate consequence of Lemma 2. Lemma 3. Let f be a weighted perfect code with fourier coecients A d, d D. Let p be a word of Z n 2 and let g be the code f+p. If g has the fourier coecients B d, d D, then Ad if p d; B d = A d if p d: (9) Let D(f) denote the set D(f) =d D A d 0}; where the numbers A d, d D, are the fourier coecients of the code f. Proposition 2. For any weighted perfect code f ker(f)= D(f) : Proof. From (9), if p ker(f) andd D with d p, thenthe fourier coecient A d of f equals zero. Hence, ker(f) D(f) : If p D(f) then, for every d D(f), p d and hence, from Lemma 3, the fourier coecients of f and f+p equals. This implies, by Proposition, that f= f+p,

6 484 O. Heden / Discrete Mathematics 256 (2002) i.e. p ker(f). Consequently, D(f) ker(f) and the proposition is proved. As for any d D, w(d) is an even number, we thus get the following generalization of a result of Shapiro and Slotnik [5]: Corollary. The kernel of a weighted perfect code f always contains the all one word (; ;:::;). 3. Main results The theorem of Avgustinovich [] is easily generalized to weighted perfect codes. The same proof will give the following theorem. Theorem 2. Any weighted perfect code f is uniquely determined by its restriction to the words of weight (n +)=2. Proof. As the all-one word (; ;:::;) is a period of f, see the corollary of Proposition 2, we get that the values f(v), where w(v)=(n )=2, are completely determined by the restrictionof f to words of weight (n +)=2. Assume that f and g are weighted perfect codes with f(v)=g(v) if w(v)=(n +)=2: We dene a function h from Z2 n to the real numbers by f(v) if w(v) 6 (n )=2; h(v)= g(v) if w(v) (n +)=2: As no -sphere S (v) cancontainboth words of weight (n )=2 and words of weight (n +)=2 +, we get that h is a weighted perfect code. From the corollary of Proposition 2, applied to the code h, we immediately get that f(v)=g(v) for all v Z. We now prove the main result. Let ; 2 ;:::; t denote the elements of D. We dene a t t matrix E =(e ij ) inthe following way: if i j ; e ij = if i j : Let i =(d ;d 2 ;:::;d n)and j =(d ;d 2 ;:::;d n ). We note that e ij is the coecient of the monomial x d xd 2 2 :::xd n n inthe polynomial n ( + x i) di ( x i ) di.

7 O. Heden / Discrete Mathematics 256 (2002) From Proposition we get that the following system of linear equations must hold for any weighted perfect code f: f( A ) S (0). +E A 2 f(. = 2 ) : (0). A t f( t ) By Proposition, any solution A ;A 2 ;:::;A t to this system of t equations will give a weighted perfect code. By the Theorem 2, this code is uniquely determined by the values f( );:::;f( t ). The fourier coecients are, by Proposition, uniquely determined by the code f. Hence there can, to any right-hand member of (0), be at most one solution of the system of equations (0). We may conclude that the matrix E must be non-singular. For any code f with a known set of values f(v), w(v)=(n +)=2, we may thus from the system of linear equations (0) uniquely calculate the fourier coecients of f, either by Gauss elimination s, taking the inverse of the matrix E or very explicitly by Cramers rule. Whenwe have got these fourier coecients we get the code from the formula inproposition. Let us also nally remark that fourier coecients and weighted perfect codes also have been considered by Anastasia Vasileva [7], but for another purpose and in another setting. Remark. It might be possible to prove directly that the matrix E above is non singular. However, we found it not quite out of interest to follow the rout via the generalization of the Shapiro-Slotnik theorem. Acknowledgements I am grateful to Faina I. Solov eva and Sergey V. Avgustinovich for mentioning this problem. References [] S.V. Avgustinovich, On a property of perfect binary codes, Discrete Anal. Oper. Res. 2() (995) 4 6. [2] O. Heden, A generalized Lloyd theorem and mixed perfect codes, Math. Scand. 37 (975) [3] F. Hergert, Algebraische Methoden fur nichtlineare Codes, Dissertion, Darmstadt, 985. [4] J-E. Roos, An algebraic study of group and nongroup error-correcting codes, Inform. Control 8 (965) [5] H.S. Shapiro, D.S. Slotnik, On the mathematical theory of error correcting codes, IBM J. Res. Develop. 3 (959) [6] F.I. Solov eva, Perfect binary codes: bounds and properties, Discrete Math. 23 (2000) [7] F.I. Solov eva, private communication.