We construct new $(v,4,1)$ binary cyclically permutable constant weight (CPCW) codes with lengths $v\le 136$ . We establish that (due to a software error) $85\,285$ ( $0.0044\%$ ) of all the $1\,939\,771\,399$ binary $(v,4,1)$ CPCW codes with $v\le 76$ were not obtained in our paper “Classification of Optimal $(v,4,1)$ Binary Cyclically Permutable Constant Weight Codes and Cyclic $2$ - $(v,4,1)$ Designs with $v\le 76$ ”, Probl. Inf. Transm., 2011, vol. 47, no. 3, pp. 224–231. We now correct the number of codes for lengths $41$ , $46$ , $53$ , $58$ , $71$ , $73$ , and $74$ , wrongly given in our 2011 paper, and add the previously missing $85\,285$ codes to the results available online. Our attempt to classify $(v,4,1)$ binary CPCW codes with $v>76$ leads to an extremely big number of codes, which makes the classification practically unusable. When $v=12n+1$ , the number of codes is relatively smaller, and they correspond to $(v,4,1)$ cyclic difference families and to cyclic $2$ - $(v,4,1)$ designs, which have various other relations and applications. Cyclic designs with one short orbit can be constructed for $v=12n+4$ . By a parallel algorithm run on the Bulgarian high performance computer Avitohol, we obtain classification results for optimal $(85,4,1)$ CPCW codes (cyclic $2$ - $(85,4,1)$ designs) and for cyclic $2$ - $(88,4,1)$ designs. For all other $76<v\le 136$ , we do not construct all $(v,4,1)$ CPCW codes but provide files with thousands of new optimal codes such that each of them differs in many codewords from most of the other codes.
