It is indeed a fundamental construction of all linear codes

Abstract

Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the generator matrix approach is called a fundamental construction of linear codes. This is the only known construction method that can produce all linear codes over finite fields. Recently, a defining-set construction of linear codes over finite fields has attracted a lot of attention, and have been employed to produce a huge number of classes of linear codes over finite fields. It was claimed that this approach can also generate all linear codes over finite fields. But so far, no proof of this claim is given in the literature. The objective of this paper is to prove this claim, and confirm that the defining-set approach is indeed a fundamental approach to constructing all linear codes over finite fields. As a byproduct, a trace representation of all linear codes over finite fields is presented.