Proves h_q(n)=0 iff every nonzero cyclic code has a full-weight codeword, gives sharp bounds and examples for positive h_q(n), determines counts over F_2 with lower bounds over F_3, and shows h_q((q^m+1)/2)>0 for odd prime q>=3 and m>=4.
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Cyclic Codes and Cyclically Covering Subspaces over Finite Fields
Proves h_q(n)=0 iff every nonzero cyclic code has a full-weight codeword, gives sharp bounds and examples for positive h_q(n), determines counts over F_2 with lower bounds over F_3, and shows h_q((q^m+1)/2)>0 for odd prime q>=3 and m>=4.