Research On Type â…¡ Codes And Gray Map Over Finite Rings

Type Ⅱ code is an special class of self-dual codes and there is a connection between codes over finite rings and codes over finite fields through Gray maps．In this paper, we mainly study Type Ⅱ codes over F2＋uF2＋…＋ukF2and the Gray image of (1＋u)-constacyclic codes over F2m＋uF2m＋vF2m＋vF2m.·The details are given as follows：(1)A method of construction self-orthogonal and self-dual code of any even length over F2＋uF2＋…＋ukF2is given in this paper. Euclidean weight of each element in F2＋uF2＋…＋ukF2is defined and self-dual codes over F2＋uF2＋…＋ukF2are proved to be Type Ⅱ codes with Euclidean weights that are a multiple of2k+2．(2) We introduce a Gray map φ from F2m＋uF2m＋vF2m＋uvF2m to F2m4.It is proved that the image of (I＋u）-constacyclic codes over F2m＋uF2m＋VF2m＋uvF2m of arbitrary length n under the Gray mapφis a distance-invariant quasi-cyclic codes of index2and length4n over F2m..It is also proved that the Gray image of cyclic codes of odd length n under such Gray map is permution-equivalent to quasi-cyclic codes of index2and length4n over F2m．Some optimal binary codes are also obtained·(3)We introduce a Gray map φhom. from F2m＋uF2m＋vF2m＋uvFm to F2m8.It is proved that the image of (I＋u)-constacyclic codes over F2m＋uF2m＋vF2m＋uvF2m of arbitrary length n under the Gray mapφhom is a distance-invariant quasi-cyclic codes of index4and length8n over F2m..It is also proved that the Gray image of cyclic codes of odd length n under such Gray map is permution-equivalent to quasi-cyclic codes of index4and length8n over F2m.Some optimal binary codes are also obtained．...