Calculating length of code words in quantum information(compression)

I was studying this article by Boestrom and Felbinger.

We define the significant length of the codewords in the preparation of the communication protocol : $$L_c(w_i) = lceil log_k(i) rceil$$

We can also calculate the base length of the codewords :
$$ underline L _c(x)= max_{i=1,…,d} {L_c(w_i) | |langle w_i|xrangle|^2 > 0 } tag{101}$$

After explaining the protocol, they give an explicit example with a source message set :
$$ chi = {|arangle,|brangle, |crangle, |drangle, |erangle, |frangle, |grangle, |hrangle, |irangle, |jrangle }$$

The Authors then calculate that :
$$underline L _c (a) = 0 ; underline L _c (b) = underline L _c (c) = underline L _c (d) = 1 tag{136}$$
$$ underline L _c (e) = underline L _c (f) = underline L _c (g) = underline L _c (h) = underline L _c (i) = underline L _c (j) =2 tag{137}$$

I have troubles understanding the equation 101 and how it calculates the length of the strings. Maybe explaining the examples in equations 136 and 137, will help too.

Thanks for reading, hope you can help !