- ISlT 2000, Sorrento, Italy, June 25-30,2000 Minimal Tail-Biting Trellises for Linear MIX Codes over Fpm B.Sundar Rajan’ Department of Electrical Communication Engineering Indian Institute of Science Bangalore 560 012, India bsrajanQece.iisc.ernet.in Abstract - For all linear (n,k,d) MDS over finite fields Fpm, we identity a generator matrix with the property that the product of trellises of rows of the generator matrix will give a minimal tail-biting linear trellis, and viewing the code as a group code, identify a set of generators, product of whose trellises will give a minimal tail biting group trellis. We also give the necessary and sufficient condition for the existence of flat minimal linear and group tail-biting trellises. I. INTRODUCTION Trellis representation of block codes illuminate the structure of the code and also useful for efficient decoding. Recently, unconventional ” Tail-biting trellises” (TBT) have been stud- ied for well known codes like (24,12,8) Golay code, hexacode and few other short codes [l]. Minimal Tail-Biting Trellis: A tail-biting trellis with min- imum maximum number of states along with the minimum product of all state space sizes, among all tail-biting trellis- es for the code under all possible coordinate permutations is called a minimal tail-biting trellis for the code. The total span bound: [l] If C is an (n, k, d) linear code over Fp, then any n-section linear tail-biting trellis for C sat- isfies j=O s, 2 If g = pm, then for group trellises we have (3) Flat Trellis: A tail-biting trellis is said to be flat if it has a constant state complexity profile. It is well known that any k coordinates of a MDS code can be taken as information positions. This means that minimum weight vectors (of weight n - k + 1) with circular span n - k can be obtained such that the successive n - k + 1 nonzero components start from any specified coordinate position from {0,1, ..., (n - 1)). It can be shown that any k such vectors starting from different coordinate positions will constitute a generator matrix for the code. Using these results in the next section we specify the generator matrices that give minimal tail-biting trellises in terms of these k coordinate positions. 11. MINIMAL CIRCULAR SPAN GENERATOR MATRICES Theorem 1: For a (n, k) linear MDS code over Fp, let e = gcd(n, k), n’ = : , k’ = a and n’ = ak‘ + D, where a and ‘This work was partly supported by CSIR, India, through Re- search Grants (N0:25(0086)/97/EMRI-I1) and (22(0298)/99/EMR- 11) to B.S.Rajan . G.Viswanath Department of Electrical Communication Engineering Indian Institute of Science Bangalore 560 012, India gvisuaQprotocol.ece.iisc.ernet.in ,Ll are integers. The generator matrix which has only minimum weight vectors with consecutive nonzeros and with nonzeros starting from the indices given by the set I given below gives a minimal linear tail-biting trellis when product of trellises corresponding to each row vector is obtained: I = {[{jn‘ + i(a + 1)li = O,l,. . . ,p} U {jn’+p(a+l)+(i-p)ali =p+1, ..., k’- l}] j = 0,1, ..., (e - I).} (4) Theorem 2: A necessary and sufficient condition for an (n, k) linear MDS code over any finite field to admit a minimal linear flat-trellis is that ”n divides k’”. Notice that the condition in Theorem 2 is independent of the size of the field. Theorem 3: For a (n,k) linear MDS code over Fpm, let e = gcd(n,mk), n‘ = a, k’ = F. Also, let k’ = an’ + k“ where 0 5 k” < n‘ and n‘ = ak” + p, where 0 5 p < k” and a and a are integers. The group-generator matrix which has a + 1 minimum weight vectors with consecutive nonze- ros with nonzeros starting from the indices given by the set I given below and a minimum weight vectors with consecutive nonzeros with nonzeros starting at all other time indices gives a minimal group tail-biting trellis when product of trellises corresponding to each row vector is obtained, if the rows s- tarting at the same index are plinearly independent (which can always be achieved): I= {[{jn’+i(a+l)Ji=O,l, ...,p} U {jn’ + ~(CX + 1) + (i - p)ali p + 1,. . . . k” - l}] j = O,l, ...,( e - I).} (5) Theorem 4: A necessary and sufficient condition for a linear (n, k) MDS code over Fpm to admit a minimal group flat-trellis is that ”n divides mk’”. Observe that the condition in Theorem 4 depends only on m and not on the characteristic of the field. ACKNOWLEDGMENTS B.S.Rajan gratefully acknowledges IBM India Research Lab, for the travel support to present this paper. REFERENCES [l] A.R.Calderbank, G.D.Forney and A.Vardy, ”Minimal Tail- Biting Trellises: The Golay Code and More”, IEEE Tkans. on Information Theory, Vo1.45, No.5, pp.1435-1455, July 1999. [2] F.R.Kschischang and V.Sorokine, ”On the trellis structure of Block codes ”, IEEE IFans. Information Theory, Vo1.41, pp.1924-1937, NOV. 1995. 118 0-7803-5857-O/OO/$l O.OO 02000 IEEE.