On the length of a minimal alphabetical bonding in linear regular languages

In the Ph.D. thesis [1] it has been found the upper bound on the minimal length of two words in regular language with similar image under alphabetic coding (if such pair of words exists at all). In this paper, we investigate the problem of finding corresponding lower bounds in subcase when regular languages have a linear growth function, and the coding scheme transforms all the letters of the input alphabet into the same symbol. For such encoding, the image of any word is uniquely determined by its length. Below we give lower bounds that coincide in order with the upper bounds from [1] for such languages and such coding. In addition, a more accurate upper estimate is given for this subcase.