On a Class of Left Metacyclic Codes

Let G
_(m,3,r)
= (x, y | x
^m
= 1, y
³
= 1, yx = x
^r
y) be a metacyclic group of order 3m, where gcd(m, r) = 1, 1 <; r <; m and r
³
≡ 1 (mod m). Then, left ideals of the group algebra F
_q
[G
_(m,3,r)
] are called left metacyclic codes over Fq of length 3m, and abbreviated as left G
_(m,3,r)
-codes. A system theory for left G
_(m,3,r)
-codes is developed for the case of gcd(m, q) = 1 and r ≡ q
^E
(mod m) for some positive integer ε, only using finite field theory and basic theory of cyclic codes and skew cyclic codes. The fact that any left G(m,3 1)-code is a direct sum of concatenated codes with inner codes λ
_i
and outer codes C
_i
is proved, where . A
_i
is a minimal cyclic code over F
_q
of length m and C
_i
is a skew cyclic code of length 3 over an extension field of F
_q
. Then, an explicit expression for each outer code in any concatenated code is provided. Moreover, the dual code of each left G
_(m,3,r)
-code is given and self-orthogonal left G
_(m,3,r)
-codes are determined.