Mathematics 502
A. Hulpke
Homework (due Mar 28)
27) Write down a check matrix for the ternary Hamming code of length 13. Using this:
a) decode the received word (1,2,1,0,2, 1,0,0,1,0,2,1,0).
b) construct a generator matrix for the code.
28) Show that the design whose blocks are the supports of words of minimum weight
in the q-ary Hamming code of length (qd − 1)/(q − 1) is isomorphic to the design whose
blocks are the collinear triples of points in the projective space PG(d - 1, q).
29) a) Prove that, if C is a linear MDS code, then C ⊥ is also MDS.
b) Show that the code C corresponding to a set S of points in PG(d − 1, q) (as in theorem
17.7.1) is MDS if and only if S has the property that no d of its points are contained in a
hyperplane. (Such a set is called an arc.) Deduce that conics in PG(2, q) give rise to MDS
codes.
30) Given a linear code C ⊂ Fn we form a new set Cx ⊂ Fn+1 by adding a parity check
symbol:
(
)
n
Cx := (c0 , c1 , . . . , cn ) ∈ Fn+1 | (c1 , . . . , cn ) ∈ C , ∑ ci = 0 .
i=0
Cx is called the extended code.
a) Show that if C is a code with odd minimum distance d then Cx is a code with minimum
distance d + 1.
b) Let C the binary hamming code of length 7 and Cx the corresponding extended code.
Show that the fourteen words of weight 4 in Cx form a 3 − (8, 4, 1)-design.