To the Optimization of Shortened Hamming Codes

Optimization is treated as reduction of the number of words of weight $d$, where $d$ is the code distance. For $d=3$, the authors propose methods of synthesizing codes of arbitrary length with minimum number of words of weight 3. For extended codes with $d=4$, asymptotically coincident upper (guaranteed attainable) and lower bounds are constructed for the minimum number of words of weight 4. Classes of codes that attain the lower bound or are close to it are indicated.