TC10 / Problems 46‐60 S. Xambó , prove that the minimum distance of 46. If is a primitive element of 16 is 21 and that its dimension is 18. , / and a 1
47. Let primitive element of . Find the dimension and a control matrix (over ) 5 . Show that the minimum distance of this code is 7. of 48. Cyclotomic polynomials. In the factorization of ∏
2. 1
∏
|
, let ,
(the degree of this polynomial is polynomial over ). Prove that: 1. 1
and it is called the ‐th cyclotomic . for all , where is the characteristic of . 3. ∏
|
1
, where is the Möbius function (
is 0 if has repeated prime factors, and otherwise it is 1 or 1 according to whether the number of prime factors is even or odd). 4. If is prime, 1 and and gcd ,
1, then is the product of 5. If tinct irreducible polynomials of degree . . / dis‐
49. Let and be the irreducible factors of degree 5 of 1 over . Show that the exponents of the roots of (of ) are the quadratic is a quadratic non‐residues (the quadratic residues) modulo 11. If is the permutation of non‐residue (
2,6,7,8,10 ) and 0,1, … ,10 , prove that . This is an example of two dis‐
tinct cyclic codes that are equivalent. 50. Let be a cyclic linear binary code of odd length . Prove that con‐
tains a vector of odd weight if and only if it contains the vector . If this condition is satisfied, prove that the subcode of formed by the even‐
weight vectors is a cyclic code. 51. Find the weight enumerators of the ternary Golay codes Γ and Γ . 52. Find the dimension and minimum distance of all the binary strict BCH codes of length 15. Ditto of length 31. Ditto of length 63. 53. Calculate a control matrix of a binary BCH code of length 31 that cor‐
rects 2 errors. 54. Find a generator polynomial of a binary BCH code of length 11 and de‐
sign distance 5. 55. Compute the dimension of a BCH code over rects 5 errors. of length 80 that cor‐
56. Calculate the generator polynomial of a ternary BCH code of length 26 and design distance 5. Find also its dimension. 57. Let 7 mod 17
7,3 is 7. |
58. Let . Prove that the minimum distance of , where is the Paley matrix of (is the 5
5 matrix with 0’s along its diagonal and with 1 or 1 at the , entry according to whether 1,4 or 2,3 ). Prove that is a ternary Golay code (that is, 11,6,5 ). 59. If ,…,
, ,
is a non‐zero vector of the kernel of the matrix ,…,
diag , … ,
, prove that the dual of , , is . and 60. (Alternative definition of classical Goppa codes). Let , a positive integer. Let be a polynomial of degree 0 for all . ,…,
the elements such that 0 and Set, according to Goppa’s original definition, Γ
,
|∑
0 mod
. [ ] ,
In this problem we will see that Γ
1) Given and that Γ
,
. 0, show that such that is invertible modulo mod (note that is a polynomial of degree with coefficients in ). 2) Show that the condition ∑
0 mod
is equivalent to ∑
0. 3) Use this relation to prove that the code defined by [ ] admits a control matrix of the form ,…,
diag
,…,
, where 0 si 1/
, 0 o (here ,
, with the convention that ). 4) Since is invertible, the code defined by [ ] also admits a control matrix of the form ,…,
and this establishes Γ
Γ , . diag
,
,…,
Γ
,
, , as is a control matrix for