tor polynomials of the self-orthogonal binary cyclic codes of lengths 7,9 , and 15 . Compare your answers to those of Exercise 252 . Exercise 257 Let j(x)=1+x+x2++xn1 in Rn and j(x)=(1)j(x). In Exercise 221 we gave properties of j(x) and j(x). Let C be a cyclic code over Fq with generating idempotent i(x). Let Ce be the subcode of all even-like vectors in C. In Exercise 238 we found the generator polynomial of Ce. (a) Prove that 1j(x) is the generating idempotent of the [n,n1] cyclic code over Fq consisting of all even-like vectors in Rn. (b) Prove that i(1)=0 if C=Ce and i(1)=1 if C=Ce. (c) Prove that if C=Ce, then i(x)j(x) is the generating idempotent of Ce. Exercise 238 Let C be a cyclic code over Fq with defining set T and generator polynomial g(x). Let Ce be the subcode of all even-like vectors in C. (a) Prove that Ce is cyclic and has defining set T{0}. (b) Prove that C=Ce if and only if 0T if and only if g(1)=0. (a) Prove that j(x)2=nj(x) in Rn. (b) Prove that j(x) is an idempotent in Rn. (c) Prove that j(x) is the generating idempotent of the repetition code of length n over Fq. (d) Prove that if c(x) is in Rn, then c(x)j(x)=c(1)j(x) in Rn. (e) Prove that if c(x) is in Rn, then c(x)j(x)=0 in Rn if c(x) corresponds to an even-like vector in Fqn and c(x)j(x) is a nonzero multiple of j(x) in Rn if c(x) corresponds to an odd-like vector in Fqn. tor polynomials of the self-orthogonal binary cyclic codes of lengths 7,9 , and 15 . Compare your answers to those of Exercise 252 . Exercise 257 Let j(x)=1+x+x2++xn1 in Rn and j(x)=(1)j(x). In Exercise 221 we gave properties of j(x) and j(x). Let C be a cyclic code over Fq with generating idempotent i(x). Let Ce be the subcode of all even-like vectors in C. In Exercise 238 we found the generator polynomial of Ce. (a) Prove that 1j(x) is the generating idempotent of the [n,n1] cyclic code over Fq consisting of all even-like vectors in Rn. (b) Prove that i(1)=0 if C=Ce and i(1)=1 if C=Ce. (c) Prove that if C=Ce, then i(x)j(x) is the generating idempotent of Ce. Exercise 238 Let C be a cyclic code over Fq with defining set T and generator polynomial g(x). Let Ce be the subcode of all even-like vectors in C. (a) Prove that Ce is cyclic and has defining set T{0}. (b) Prove that C=Ce if and only if 0T if and only if g(1)=0. (a) Prove that j(x)2=nj(x) in Rn. (b) Prove that j(x) is an idempotent in Rn. (c) Prove that j(x) is the generating idempotent of the repetition code of length n over Fq. (d) Prove that if c(x) is in Rn, then c(x)j(x)=c(1)j(x) in Rn. (e) Prove that if c(x) is in Rn, then c(x)j(x)=0 in Rn if c(x) corresponds to an even-like vector in Fqn and c(x)j(x) is a nonzero multiple of j(x) in Rn if c(x) corresponds to an odd-like vector in Fqn