Multiplicative Groups Used For Coding Three Dimensional Signals

Shannon has illustrated the method for reliable communication in noisy channels,which marks the beginning of coding theory.H μber use the algebraic integer ring of quadratic cyclotomic field to construct block codes for coding two dimensional signals which are able to correct some error patterns.In l998,DongXuedong etal.used the algebraic integer ring of unique factorization to construct block codes for coding even number dimensional signals which are able to correct some error. In this thesis,we use the algebraic integer ring of the algebraic number field Q(-2^1/3) to obtain three dimensional signal spaces and to construct block codes for coding three dimensional signals which are able to correct some error patterns.Firstly,we define a Z-module isomorphism from Z[θ] onto Z³,this is a one-to-one correspondence between the elements [a,b,c](a + bθ + cθ²) in z[θ] and the integral points in the three dimensional space Z³.Next, the multiplicative groups of units in the quotient rings Z[θ]/(2ⁿ) and Z[θ]/(pⁿ) are respectively denoted by G_n² and G_n^p,where Z[-2^1/3] is the algebraic integer ring of the algebraic number field Q(-2^1/3) and p is a given odd prime number. These multiplicative groups of units can be factored as direct products of cyclic groups.For a given n,the images of the complete sets of coset representatives of the groups G_n² and G_n^p under (?) can be used as three dimensional signal spaces.Finally,we use the groups G_n² and H_n^p,which is a subgroup with 2p^3n-3 elements of G_n^p,as a tool to construct error-correcting codes for coding three dimensional signals.Then,code C_I associated with the group G_n² or H_n^p and error patterns corrected have been given.