The Depth Spectrum Of Linear Codes Over Some Special Rings And Related Problems Of Number Theory

Let q be a power of the prime.By generalizing the concept for the depth of vectors from F_qto the ring R=F₃+v F₃（v²=1）,a recursive algorithm for the depth of the vector over R is obtained.Furthermore,the upper bound and lower bound for the depth spectra of linear codes with an arbitrary code length over R are obtained.Let α≧2 be an integer,p₁and p₂be odd prime numbers with p₁<p₂.By using ele-mentary methods and techniques,we improve the relevant results about near-perfect numbers of the form 2^α-1p₁²p₂² with other redundant divisors.Furthermore,for a fixed positive inte-ger k≧2,by generalizing the definition of near-perfect numbers to be k-weakly-near-perfect numbers,we prove that there is no k-weakly-near-perfect numbers of the form 2^α-1p₁²p₂².For any positive integer n,let σ（n）be the sum of all the positive divisors of n.We prove that the equation σ（x³）=y²has no positive integer solutions（x,y）with the form x=3p^r,where p≧5 is an odd prime,and r>1 is a positive integer.