Three Parameters in Software

Creating ANSI/AIM Code 39 in Software Three Parameters

Three Parameters
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Before we continue with our discussion, we need to mention that any coding scheme needs to have at least three parameters: the codeword size n, the dataword size k, and the minimum Hamming distance dmin A coding scheme C is written as C(n, k) with a separate expression for dmin- For example, we can call our first coding scheme C(3, 2) with d min = 2 and our second coding scheme C(5, 2) with d min ::= 3
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Hamming Distance and Error
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Before we explore the criteria for error detection or correction, let us discuss the relationship between the Hamming distance and errors occurring during transmission When a codeword is corrupted during transmission, the Hamming distance between the sent and received codewords is the number of bits affected by the error In other words, the Hamming distance between the received codeword and the sent codeword is the number of bits that are corrupted during transmission For example, if the codeword 00000 is sent and 01101 is received, 3 bits are in error and the Hamming distance between the two is d(OOOOO, 01101) =3
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Minimum Distance for Error Detection
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Now let us find the minimum Hamming distance in a code if we want to be able to detect up to s errors If s errors occur during transmission, the Hamming distance between the sent codeword and received codeword is s If our code is to detect up to s errors, the minimum distance between the valid codes must be s + 1, so that the received codeword does not match a valid codeword In other words, if the minimum distance between all valid codewords is s + 1, the received codeword cannot be erroneously mistaken for another codeword The distances are not enough (s + 1) for the receiver to accept it as valid The error will be detected We need to clarify a point here: Although a code with d min = s + 1
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ERROR DETECTION AND CORRECTION
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may be able to detect more than s errors in some special cases, only s or fewer errors are guaranteed to be detected
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To guarantee the detection of up to s errors in all cases, the minimum Hamming distance in a block code must be d min S + 1
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Example 107
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The minimum Hamming distance for our first code scheme (Table 101) is 2 This code guarantees detection of only a single error For example, if the third codeword (l0 1) is sent and one error occurs, the received codeword does not match any valid codeword If two errors occur, however, the received codeword may match a valid codeword and the errors are not detected
Example 108
Our second block code scheme (Table 102) has dmin = 3 This code can detect up to two errors Again, we see that when any of the valid codewords is sent, two errors create a codeword which is not in the table of valid codewords The receiver cannot be fooled However, some combinations of three errors change a valid codeword to another valid codeword The receiver accepts the received codeword and the errors are undetected
We can look at this geometrically Let us assume that the sent codeword x is at the center of a circle with radius s All other received codewords that are created by 1 to s errors are points inside the circle or on the perimeter of the circle All other valid codewords must be outside the circle, as shown in Figure 108 Figure 108 Geometric concept for finding dmin in error detection
I I I
Legend Any valid codeword Any corrupted codeword with 0 to s errors
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In Figure 108, dmin must be an integer greater than s; that is, d min
=s + 1
Minimum Distance for Error Correction
Error correction is more complex than error detection; a decision is involved When a received codeword is not a valid codeword, the receiver needs to decide which valid codeword was actually sent The decision is based on the concept of territory, an exclusive area surrounding the codeword Each valid codeword has its own territory We use a geometric approach to define each territory We assume that each valid codeword has a circular territory with a radius of t and that the valid codeword is at the
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