Covering Sets Of Residue Class Ring

Covering codes and covering sets play an important role in coding theory,and covering codes have important applications in rewriting flash memories.In order to improve the density of flash memories,muti-level(q-level)memory cells are used so that each cell stores log2(q)bits.Even though muti-level cells increase the storage density comparing to single-level cells,they also impose two important challenges.The first one is that the voltage difference between the states is narrowed since the maximum voltage is limited.A natural consequence is that reliability issues such as low data retention and read/write disturbs become more significant.The second major challenge in flash memory systems is that the writing mechanism is relatively time consuming.However the minimum covering set for some parameters can improve the speed of the rewriting flash memory.For these reasons,studying covering sets and covering codes has extremely important theoretical and practical significance.In recent years,number theory and finite field are applied to study the minimum covering set for some parameters.Meanwhile,bounds on the parameters of some covering codes have also been obtained.In this thesis,we also utilize number theory and finite ring to study covering sets for limited-magnitude errors.Let integers A,r,q with r | q and gcd(r,6)= 1.In the first part we introduce a result reducing the construction of a minimum size(3,0;3lr)-covering set for k ? 2 to the construction of a minimum size(3,0;3l-2r)-covering set.In the second part we study the minimum size(3,0;3lr)-covering set and the minimum(4,0;3lr)-covering set for l ? 2 in a similar way.