Quasi-Perfect Linear Codes With Minimum Distance $4$

Abstract

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n,n-m,4] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> -code when either 1) q=16, m ges 5, m odd, or 2) q=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , 7 les i les 15, m ges 4, or 3) q=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2lscr </sup> , lscr ges 8, m ges 5, m odd. As quasi-perfect [n,n-m,4] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> -codes and complete n-caps in projective spaces PG(m-1,q) are equivalent objects, new upper bounds on the size of the smallest complete cap in PG(m-1,q) are obtained