インターネットチェックサムの2重, 3重誤り検出能力

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The checksum procedure for TCP or IP is formulated as the 16-bit one's complement addition of ten or more words. Let w_i denote the i-th 16-bit word in a transmitted data. The sixth word w_6 denotes the checksum part. When the transmitted packet consists of n >__- 10 words, the checksum is calculated as follows. w_6 = <w_1+'...+'w_5+'w_7+'...+'w_n.>^^^- (1) where +' denotes one's complement addition. Let w'_i denote the i-th word in a received packet. To check the checksum, c = w'_1+'...+'w'_n is computed. If and only if |c|_<(1)> = -0, we decide that the check succeeds. For two integers m >_- 3, n >_- 3, consider the vector u = (u_0. u_1.....u_<mn-1>) of length mn. Let u_i = (u_<mi-> u_<mi+1>.....u_<mi+m-1>) for 0 <_- i <_- n-1. We write u = (u_0, u_1....u_<n-1>). Let C_<m,n> be the following nonlinear code of length mn. C_<m,n> = {u : u_<n-1> = <u_0+'u_1+'...+'u_<n-2>}.^^^- (2) When m = 16 and n >_- 10, C_<m,n> is the instance for the checksum procedure in TCP and IP. For two vectors u = (u_0, u_1.....u_<m-1>) and v = (v_0, v_1.....v_<m-1>), the Hamming distance d(u,v) is the number of coordinate i for which u_i ≠ v_i. For a code C of length, N and 0 <_- i <_- N, B_i is defined as follows. [numerical formula] (3) B_0, B_1....B_N is called the distance distribution of C. Suppose that we use the code C only for error detection in a binary symmetric channel with bit error rate e. Assume that each codeword is transmitted with the same probability. Then, the probability of an undetectable error P_<ue> is given as follows. [numerical formula]. (4) In the next section, we examine a part of the distance distribution of C_<m,n> by a combination approach.
社団法人電子情報通信学会の論文
1997-08-13

インターネットチェックサムの2重, 3重誤り検出能力

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