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p53 genes have something to do with 50% tumor of mankind, Liver cancer、breast cancer、urine bladder cancer、stomach cancer、colon cancer、prostate gland cancer、soft organization tumor、ovary cancer、brain lump、lymphoid cell tumor、esophagus cancer、lung cancer、ossification tumor and so on have been found currently. p53 in human tumors mainly mutates in the highly conservative area, and mutation frequency at 175、248、249、273、282 locus. p53 gene as a Target for gene therapy is always a hotspot for tumor treatment research.Main contributions of our work mainly include the several aspects below:1、In the alphabet {A,C,G,T},the following bases are introduced ’D’: others ’P’:{A,T}→{G,C};’O’:{G ,C}→{A ,T};So, elements of finite field C343 can be expressed X1 X 2 X 3, where X i∈{D,A,C,O,G,T,P}, the following bijectionf:GF（7）→{ D,A,C,O,G,T,P} , f（0）=D,f（1） =A, f（2）=C,f（3）=O,f（4）=G,f（5）=T,f（6）=Pand mapping * : GF（73）→C343, that isφ（a0+a1x+a2x2）=(f（a1）f（a2）f（a0） =（X1X2X3）furthermore multiplication, addition and scalar over finite field C343 are defined. Theorems based on this structure are applied in a few kinds of cancer genes, so we may presage new possible pathology gene sequences to prevent disease.2、Definitions of bran-new distance and Y-inner product can be given on the base of the above algebraic structure. By calculation, 36 cubic primitive polynomials and their roots on GF（7） can be got, and the polynomial and its root which has the most biology significance can also be got. Then, during the p53 mutation in stomach cancer, esophagus cancer, lung cancer and B cell lymphoma, the hydrophobicity of amino acid are studied by using the theoretical models, and gets some interesting results.3、A new algebraic structure can be established on the base of the above algebraic structure, and based on which, hamming distancedH（X1Y1Z1,X2Y2Z2）can be defined. Improving hamming distance, H*- distance has been got. Then, the cross product between codons is defined. Finally, in accordance with hamming distance, H*- distance and the cross product, H * - inner product between codons is defined, and studies a few kinds of cancer genes by using above hamming distance, H*- distance, the cross product and H*- inner product. We find the latter is more meaningful for our study by comparing hamming distance with H*- distance. The innovations of this paper are:1、Establish an oncogene algebraic structure.2、Analyze a few kinds of cancer genes using theorems based on this structure.3、Extend and perfect algebraic structure, in turn new H*-distance and H*-inner product are defined, and analyze a few kinds of cancer genes.
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