Optimal Equi-difference Conflict-avoiding Codes
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An equi-differece conflict-avoiding code (CAC^e) C of length n and weight ω is a collection of ω-subsets (called codewords) which has the form {0,i,2i,...,(ω-1)i} of Z_n such that Δ(c_1)∩Δ(c_2)=∅ holds for any c_1,c_2∈C, c_1≠ c_2 where Δ(c)={j-i (modn) | i,j∈ c,i≠ j}. A code C∈ CAC^es with maximum code size for given n and ω is called optimal and is said to be perfect if ∪_c∈CΔ(c)=Z_n{0}. In this paper, we show how to combine a C_1∈ CAC^e(q_1,ω) and a C_2∈ CAC^e(q_2,ω) into a C∈ CAC^e(q_1q_2,ω) under certain conditions. One necessary condition for a CAC^e of length q_1q_2 and weight ω being optimal is given. We also consider explicit construction of perfect C∈ CAC^e(p,ω) of odd prime p and weight ω≥3. Finally, for positive integer k and prime p≡1 (mod 4k), we consider explicit construction of quasi-perfect C∈ CAC^e(2p,4k+1).
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