Coding Theory lecture: MDS codes

0.06 MB·English

by  KIT Tyabandha

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).
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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
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Problem 1. Showthatlinear[n;1;n℄,[n;n(cid:0)1;2℄and [n;n;1℄
odesexist overany(cid:12)nite (cid:12)eldF.
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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.
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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.
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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.
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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
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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?.
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Theorem 11. For any prime power q, we have m(2;q)=q+1.
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Theorem 12.
m(k;q)=k+1
for q (cid:20)k.
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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