On the depth distribution of linear codes

The depth distribution of a linear code was recently introduced by T. Etzion (see ibid., vol.43, pp.1361-3, July 1997). In this correspondence, a number of basic and interesting properties for the depth of finite words and the depth distribution of linear codes are obtained. In addition, we study the enumeration problem of counting the number of linear subcodes with the prescribed depth constraints, and derive some explicit and interesting enumeration formulas. Furthermore, we determine the depth distribution of Reed-Muller code RM (m,r). Finally, we show that there are exactly nine depth-equivalence classes for the ternary [11,6,5] Golay codes.