The Generalized Hamming Weights For (n,2) Linear Codes On The Field F_q And Binary (n,3) Linear Codes

Content: There are three parts in this thesis.The first part: We introduce general results about the Generalized Hamming Weights for codes ,and main results in the thesis.The second part: We give some preliminaries about codes ;including some concepts about codes and linear codes and so on.The third part: We further analyze the generator matrix of the linear codes on the field Fq , and we not only obtain the perfect numeration of the Generalized Hamming Weights for [n,2] linear codes on the field Fq , but also get the special result when q is equal to 2.Besides,we get the numeration of the Generalized Hamming Weights for binary [n,3] linear codes up to equivalence and some existence theorems .The followings are the main results:Theorem 3.1.2: The numeration of a [n,2] linear code with the Generalized Hamming Weights ( d₁ , d₂) on the field Fq :â‘ is [ d₂/ 2] + 1, if d₁ + d₃≥2d₂;â‘¡is [( d₁ + d₃ - d₂) / 2] + 1, if d₁ + d₃ < 2d₂。Theorem 3.2.1: Suppose 2d₁≤d₂ and d₃ - d₂≥d₁.The numeration of a [n,3] linear code with the Generalized Hamming Weights ( d₁ , d₂ , d3) on the field F2 :â‘ is [ d₂/ 2] + 1, if d₁ + d₃≥2d₂;â‘¡is [( d₁ + d₃ - d₂) / 2] + 1, if d₁ + d₃ < 2d₂. Theorem 3.2.2: The numeration of a [n,3] linear code with the Generalized Hamming Weights ( d₁ , d₂ , d3) on the field F2 is [( d₁ + d₃ - d₂) / 2] + 1. if 2d₁≤d₂ and d₃ - d₂ < d₁. Theorem 3.2.3: The numeration of a [n,3] linear code with the Generalized Hamming Weights ( d₁ , d₂ , d3) on the field F2 is [ d₂/ 2] + 1. if 2 d₁≥d₂≥(3d₁) / 2 and d₃ - d₂≥d₁. Theorem 3.2.7: There is not the code with the Generalized Hamming Weights (n-1-[(n-2)/3],n-1-[(n-2)/6],n),(n-[(n-2)/3],(n-[(n-2)/6],n)(n+1-[(n-2)/3],n+1-[(n-2)/6],n),…,(n+m-[(n-2)/3],n+m-[(n-2)/6,n]in the [ n, 3] linear codes. where m > [( n-2) / 6].