【作者】 李园园；

【导师】 董学东；

【作者基本信息】 辽宁师范大学 ， 应用数学， 2006， 硕士

【摘要】 Shannon阐明了在有扰信道中实现可靠通信的方法，奠定了纠错码的基石。Huber利用二次分圆域的代数整数环构作了面向二维信号且具有代数译码算法的纠错码。董学东等人利用具有唯一分解的分圆域的代数整数环构造了面向偶数维信号且具有代数译码算法的纠错码。在本论文中我们利用有理数域Q上的三次扩张Q(（-2）^1／3)的代数整数环来获得三维信号空间并构造面向三维信号且具代数译码算法的纠错码。 首先，我们定义一个从Z[θ]到Z³的Z-模同构φ，使Z[θ]中的元素[a，b，c]（表示a+bθ+cθ²）与Z³中的所有整数点（a，b，c）（a，b，c∈Z）一一对应。 其次，对于任意给定的奇素数q，把商环Z[（-2）^1／3]／（2ⁿ）和Z[（-2）^1／3]／（qⁿ）中的乘法单位群的子群G_n²和G_n^p分解成循环群的直积，在这里Z[（-2）^1／3]是代数数域Q(（-2）^1／3)的代数整数环。这样对于给定的n，G_n²及G_n^p的特定的陪集代表元的集合在φ下的象就构成了三维信号空间。 最后，利用群G_n²及G_n^p的有2p^3n-3个元素的子群H_n^p作为一个工具去构造面向三维信号能纠正单个错误类型的分组码，具体给出了与群G_n²及群H_n^p相伴的码C_I以及能纠正的错误类型。更多还原

【Abstract】 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.更多还原