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Department of Mathemati s, Mahidol University Kit Tyabandha, PhD MDS ode th 20 January 2006 Theorem 1. Given a redundan y r and a minimum distan e d. An [n;n(cid:0)r;d℄- ode satis(cid:12)es d(cid:20)r+1. x De(cid:12)nition 1. A linear [n;k;d℄ ode over F with d = n(cid:0)k +1 is alled a maximum distan e separable (MDS) ode. In other words, an MDS is a [n;n(cid:0)r;r+1℄- ode. x Theorem 2. Suppose 2(cid:20)r (cid:20)q. Let a1;:::;aq(cid:0)1 be the non-zero elements of GF(q). Then the matrix 2 3 1 1 (cid:1)(cid:1)(cid:1) 1 1 0 (cid:1)(cid:1)(cid:1) 0 66 a1 a2 (cid:1)(cid:1)(cid:1) aq(cid:0)1 0 1 (cid:1)(cid:1)(cid:1) 077 H =66 a21 a22 (cid:1)(cid:1)(cid:1) a2q(cid:0)1 0 0 ::: 077 64 ... ... ... ... ... ... ...75 r(cid:0)1 r(cid:0)1 r(cid:0)1 a1 a2 (cid:1)(cid:1)(cid:1) aq(cid:0)1 0 (cid:1)(cid:1)(cid:1) 1 is the parity he k matrix of an MDS q+1;q+1(cid:0)r;r+1 ode. Equivalently, the olumns of H form a (q+1)-ar in PG(r(cid:0)1;q). x Theorem3. LetC bealinear[n;k;d℄ odeovera(cid:12)eldF ofq elements,whereq isaprimepower with a parity he k matrix H. Then C has a ode word of weight w (cid:20)l if and only if l olumns of H are linearly dependent. x Theorem 4. Let C be a linear [n;k;d℄ ode overF with a parity he k matrix H. Then C is an MDS ode if and only if every n(cid:0)k olumns of H are linearly independent. x ? Theorem 5. If a linear [n;k;d℄ ode C is MDS, then so is its dual C . x Corollary 5[1℄. LetC bean[n;k;d℄linear odeoverF =GF(q). Thenthefollowingstatements are equivalent. a. C is MDS b. Every k olumns of a generator matrix G of C are linearly independent . Every n(cid:0)k olumns of a parity he k matrix H of C are linearly independent x Problem 1. Showthatlinear[n;1;n℄,[n;n(cid:0)1;2℄and [n;n;1℄ odesexist overany(cid:12)nite (cid:12)eldF. x De(cid:12)nition 2. We all trivial MDS odes the [n;1;n℄, [n;n(cid:0)1;2℄ and [n;n;1℄ odes. x Theorem 6. The only binary MDS odes are the trivial ones. x De(cid:12)nition 3. A squarematrixissaid to be non-singularif its olumnsarelinearlyindependent. Given any matrix A, a s(cid:2)s square submatrix of A is a s(cid:2)s matrix onsisting of the entries from some s rows and s olumn of A. x Theorem 7. Let C be an [n;k;(cid:0)(cid:0)℄ ode with parity he k matrix H =(A In(cid:0)k). Then C is an MDS ode if and only if every square submatrix of A is non-singular. x Theorem 8. Let C be an [n;k;(cid:0)(cid:0)℄ ode with generator matrix G = (Ik A). Then C is an MDS ode if and only if every square submatrix of A is non-singular. th Coding theory, MDS ode {1{ From 8 November 2005, as of 20 January, 2006 Department of Mathemati s, Mahidol University Kit Tyabandha, PhD x Theorem 9. Let C be an [n;k;d℄ MDS ode. Then any k symbols of the ode words may be taken as message symbols. x Theorem 10. Let C be an [n;k;d℄ ode over GF(q). Then C is an MDS ode if and only if C has a minimum distan e ode word with non-zero entries in any d oordinates. x Corollary 10[1℄. The number of ode words of weight n(cid:0)k+1 in an [n;k;d℄ MDS ode over GF(q) is (cid:18) (cid:19) n (q(cid:0)1) n(cid:0)k+1 x Problem 2. Given k and q, (cid:12)nd the largest value, m(k;q), of n su h that [n;k;n(cid:0)k+1℄ MDS ode exists overGF(q). x Be ause of Theorem 5, Problem 2 is equivalent to Problem 3. Problem3. Givenk andq, (cid:12)ndthelargestnforwhi hthereisak(cid:2)nmatrixoverGF(q), every k olumns of whi h are linearly independent. x Problem 4. Given a k-dimensional ve tor spa e V over GF(q), what is the order of a largest subset of V every k ve tors of whi h form a basis of the same?. x Theorem 11. For any prime power q, we have m(2;q)=q+1. x Theorem 12. m(k;q)=k+1 for q (cid:20)k. x Bibliography Raymond Hill. A (cid:12)rst ourse in oding theory. Clarendon, 1986 L R Vermani. Elements of algebrai odng theory. Chapman & Hall, 1996 th Coding theory, MDS ode {2{ From 8 November 2005, as of 20 January, 2006