| Summary: | The undetected error probability, P(e), of linear (n,k)
block codes used for transmission over a binary symmetric
channel is examined. A class of codes referred to as proper
codes is seen to possess certain desirable characteristics. A
family of increasingly better tests for determining when a code
is not proper is derived. Some known improper codes are examined
to assess the relative strengths of the tests.
For cyclic redundancy codes (CRC), P(e) is shown to be
asymptotically equal to 2[sup –p]. For specific values of k, P(e) was evaluated using the dual code weights. This greatly reduces the computational burden. The even CRC subclass consisting of generator polynomials, g(x)'s, with an even number of terms is shown to have the ability to detect all odd weight errors. It was observed that the P(e) characteristics of a code generated by g(x) is closely related to the exponent of g(x). In particular, a low exponent is a good indicator of an improper code. It was also found that by examination of a number of primitive g(x) that the resulting codes are proper.
For k < 150, a code referred to as CRC-12R is shown to have better P(e) characteristics than the CRC-12 standard. Three CRC-16 standards: CRC-ANSI, CRC-CCITT and CRC-CCIR are compared. It was shown that CRC-CCIR has a very high P(e) and CRC-CCITT has the best overall P(e) characteristics of the three standards. === Applied Science, Faculty of === Electrical and Computer Engineering, Department of === Graduate
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