# The build-up construction over a non-unital ring (List of codes up to 12)

### Abstract

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=< a,b,2a=2b=0, a^2=a, b^2=b,ab=a,ba=b>.$ We study a recursive construction of self-orthogonal codes over $E.$ We classify self orthogonal codes of length $n$ and size $2^n$ (called here quasi self-dual codes or QSD), and free rank greater than 1, up to $n=12.$ In particular, we classify Type IV codes (QSD codes with even weights) of free rank greater than $1,$ up to $n=12.$

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