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In this paper, we investigate some properties of dual automorphism invariant modules over right perfect rings. Also, we introduce the notion of dual automorphism invariant cover and prove the existence of dual automorphism invariant cover. Moreover, we give the necessary and sufficient condition for every cyclic module to be a dual automorphism invariant module over a semi perfect ring and we prove that supplemented quasi projective module has finite exchange property. Also we give a characterization of a perfect ring using dual automorphism invariant module.

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In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

A module M is said to be an automorphism liftable module if for each submodule N of M , every automorphism of the quotient M/N can be lifted to an endomorphism of M. In this work, some properties of automorphism liftable modules are investigated. Also, characterization for some special rings such as perfect, semiperfect and uniserial are given by using automorphism liftable modules.