EXERCISES CODING THEORY SHEET 1
(1) Implement the Calculation of the fiscal code control letter in Magma.
Generate random examples and estimate the probability to obtain a given letter. Is it uniform? How would you change the tables in https://en.wikipedia.org/wiki/Italian_fiscal_code_card in order to get a uniform probability distribution?
(2) Implement in Magma the 1-error detecting code obtained encoding numbers modulo an integer n (the most exploited example being n = 10) and checking that a + b = c reducing modulo n.
(3) List all elements of the original binary Hamming code C of length 7 and dimension 4.
(4) Define Supp(c) = {s : c
s6= 0}. Determine min{|Supp(c)| : c ∈ C}.
(5) Let C
minbe the set all vectors of C realizing the above defined min- imum. Let B = {Supp(c) : c ∈ C
min}. Show that B ∩ B
0contains exactly one element for B, B
0∈ B.
E-mail address: andrea.previtali@unimib.it Webpage: http://www.matapp.unimib/~prevital
Date: November 14, 2016.
Andrea Previtali. c
1