Proceedings of the 44th Symposium on Ring Theory and ...

Proposition 10. The homomorphism δ : C ⊥ → C ◦ sending x to δ x is an isomorphism
<strong>of</strong> R-modules.
Theorem 11. Let R be a finite commutative QF ring. For a submodule C ⊆ R n , (C ⊥ ) ⊥ =
C.
By Theorem 1 and Theorem 4, we can get <strong>the</strong> following corollary.
Corollary 12. Let R be a finite commutative QF ring. Then C is a polycyclic code if
and only if C ⊥ is a sequential code.
Theorem 13. Let R be a finite commutative QF ring. If C ⊆ R n is a free R-module <strong>of</strong>
finite rank, <strong>the</strong>n C ⊥ is a free R-module <strong>of</strong> rankC ⊥ = n − rankC.
We determine <strong>the</strong> parity check matrix <strong>of</strong> a constacyclic code.
Proposition 14. Let R be a finite commutative QF ring and f = X n − α ∈ R[X].
Suppose f = hg ∈ R[X] where g and h are polynomials <strong>of</strong> degree n−k and k, respectively.
Let C be <strong>the</strong> linear [n, k]-code corresponding to <strong>the</strong> ideal generated by g in R[X]/(X n − α)
and h(X) = h k X k + h k−1 X k−1 + · · · + h 1 X + h 0 . Then C has <strong>the</strong> (n − k) × n parity check
matrix H given by
⎛
⎞
h k · · · h 1 h 0 0 · · · 0
0 h k · · · h 1 h 0 · · · 0
.
H =
0 . .
. . .
. . .
. . .
. . . 0
.
⎜
⎟
⎝ .
. ⎠
0 · · · 0 h k · · · h 1 h 0
Definition 15. Let R be a finite commutative QF ring. For a sequential code C ⊆ R n , C
is called a principal sequential code if C ⊥ is a principal polycyclic code. And C is called a
principal sequential code with an invertible constant term if C ⊥ is a principal polycyclic
code with an invertible constant term.
Now we can get <strong>the</strong> main <strong>the</strong>orem.
Theorem 16. Let R be a finite commutative QF ring. Suppose C is a free codes <strong>of</strong> R n .
Then <strong>the</strong> following conditions are equivalent:
(1) Both C and C ⊥ are principal polycyclic codes with invertible constant terms.
(2) Both C and C ⊥ are principal sequential codes with invertible constant terms.
(3) C is a principal polycyclic and sequential code with an invertible constant term.
(4) C ⊥ is a principal polycyclic and sequential code with an invertible constant term.
(5) C = (g)/(X n − α) is a constacyclic code with an invertible α.
(6) C ⊥ = (q)/(X n − β) is a constacyclic code with an invertible β.
Acknowledgement. The author wishes to thank Pr<strong>of</strong>. Y. Hirano, Naruto University <strong>of</strong>
Education, for his helpful suggestions and valuable comments.
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