--- /dev/null
+/* File: golay23.c
+ * Title: Encoder/decoder for a binary (23,12,7) Golay code
+ * Author: Robert Morelos-Zaragoza (robert@spectra.eng.hawaii.edu)
+ * Date: August 1994
+ *
+ * The binary (23,12,7) Golay code is an example of a perfect code, that is,
+ * the number of syndromes equals the number of correctable error patterns.
+ * The minimum distance is 7, so all error patterns of Hamming weight up to
+ * 3 can be corrected. The total number of these error patterns is:
+ *
+ * Number of errors Number of patterns
+ * ---------------- ------------------
+ * 0 1
+ * 1 23
+ * 2 253
+ * 3 1771
+ * ----
+ * Total number of error patterns = 2048 = 2^{11} = number of syndromes
+ * --
+ * number of redundant bits -------^
+ *
+ * Because of its relatively low length (23), dimension (12) and number of
+ * redundant bits (11), the binary (23,12,7) Golay code can be encoded and
+ * decoded simply by using look-up tables. The program below uses a 16K
+ * encoding table and an 8K decoding table.
+ *
+ * For more information, suggestions, or other ideas on implementing error
+ * correcting codes, please contact me at (I'm temporarily in Japan, but
+ * below is my U.S. address):
+ *
+ * Robert Morelos-Zaragoza
+ * 770 S. Post Oak Ln. #200
+ * Houston, Texas 77056
+ *
+ * email: robert@spectra.eng.hawaii.edu
+ *
+ * Homework: Add an overall parity-check bit to get the (24,12,8)
+ * extended Golay code.
+ *
+ * COPYRIGHT NOTICE: This computer program is free for non-commercial purposes.
+ * You may implement this program for any non-commercial application. You may
+ * also implement this program for commercial purposes, provided that you
+ * obtain my written permission. Any modification of this program is covered
+ * by this copyright.
+ *
+ * == Copyright (c) 1994 Robert Morelos-Zaragoza. All rights reserved. ==
+ */
+
+#include <stdio.h>
+#include <stdlib.h>
+#define X22 0x00400000 /* vector representation of X^{22} */
+#define X11 0x00000800 /* vector representation of X^{11} */
+#define MASK12 0xfffff800 /* auxiliary vector for testing */
+#define GENPOL 0x00000c75 /* generator polinomial, g(x) */
+
+/* Global variables:
+ *
+ * pattern = error pattern, or information, or received vector
+ * encoding_table[] = encoding table
+ * decoding_table[] = decoding table
+ * data = information bits, i(x)
+ * codeword = code bits = x^{11}i(x) + (x^{11}i(x) mod g(x))
+ * numerr = number of errors = Hamming weight of error polynomial e(x)
+ * position[] = error positions in the vector representation of e(x)
+ * recd = representation of corrupted received polynomial r(x) = c(x) + e(x)
+ * decerror = number of decoding errors
+ * a[] = auxiliary array to generate correctable error patterns
+ */
+
+static int encoding_table[4096], decoding_table[2048];
+static int position[23] = { 0x00000001, 0x00000002, 0x00000004, 0x00000008,
+ 0x00000010, 0x00000020, 0x00000040, 0x00000080,
+ 0x00000100, 0x00000200, 0x00000400, 0x00000800,
+ 0x00001000, 0x00002000, 0x00004000, 0x00008000,
+ 0x00010000, 0x00020000, 0x00040000, 0x00080000,
+ 0x00100000, 0x00200000, 0x00400000 };
+static long arr2int(a,r)
+/*
+ * Convert a binary vector of Hamming weight r, and nonzero positions in
+ * array a[1]...a[r], to a long integer \sum_{i=1}^r 2^{a[i]-1}.
+ */
+int r;
+int *a;
+{
+ int i;
+ long mul, result = 0, temp;
+
+ for (i=1; i<=r; i++) {
+ mul = 1;
+ temp = a[i]-1;
+ while (temp--)
+ mul = mul << 1;
+ result += mul;
+ }
+ return(result);
+}
+
+void nextcomb(n, r, a)
+/*
+ * Calculate next r-combination of an n-set.
+ */
+int n, r;
+int *a;
+{
+ int i, j;
+
+ a[r]++;
+ if (a[r] <= n)
+ return;
+ j = r - 1;
+ while (a[j] == n - r + j)
+ j--;
+ for (i = r; i >= j; i--)
+ a[i] = a[j] + i - j + 1;
+ return;
+}
+
+long get_syndrome(pattern)
+/*
+ * Compute the syndrome corresponding to the given pattern, i.e., the
+ * remainder after dividing the pattern (when considering it as the vector
+ * representation of a polynomial) by the generator polynomial, GENPOL.
+ * In the program this pattern has several meanings: (1) pattern = infomation
+ * bits, when constructing the encoding table; (2) pattern = error pattern,
+ * when constructing the decoding table; and (3) pattern = received vector, to
+ * obtain its syndrome in decoding.
+ */
+long pattern;
+{
+ long aux = X22;
+
+ if (pattern >= X11)
+ while (pattern & MASK12) {
+ while (!(aux & pattern))
+ aux = aux >> 1;
+ pattern ^= (aux/X11) * GENPOL;
+ }
+ return(pattern);
+}
+
+static int golay23_test(int error_pattern) {
+ int data;
+ int codeword;
+ int recd;
+ int pattern;
+ int decerror;
+ int i, tests;
+
+ decerror = 0;
+ tests = 0;
+
+ for (data = 0; data<(1<<12); data++) {
+
+ codeword = encoding_table[data];
+ recd = codeword ^ error_pattern;
+ recd ^= decoding_table[get_syndrome(recd)];
+ pattern = (recd ^ codeword) >> 11;
+ for (i=0; i<12; i++)
+ if (pattern & position[i])
+ decerror++;
+ if (decerror) {
+ printf("data: 0x%x codeword: 0x%x recd: 0x%x\n", data, codeword, recd);
+ printf("there were %d decoding errors\n", decerror);
+ exit(1);
+ }
+ tests++;
+ }
+
+ return tests;
+}
+
+int main(void)
+{
+ int i;
+ long temp;
+ long pattern;
+ int tests;
+ int a[4];
+ int error_pattern;
+
+
+ /*
+ * ---------------------------------------------------------------------
+ * Generate ENCODING TABLE
+ *
+ * An entry to the table is an information vector, a 32-bit integer,
+ * whose 12 least significant positions are the information bits. The
+ * resulting value is a codeword in the (23,12,7) Golay code: A 32-bit
+ * integer whose 23 least significant bits are coded bits: Of these, the
+ * 12 most significant bits are information bits and the 11 least
+ * significant bits are redundant bits (systematic encoding).
+ * ---------------------------------------------------------------------
+ */
+ for (pattern = 0; pattern < 4096; pattern++) {
+ temp = pattern << 11; /* multiply information by X^{11} */
+ encoding_table[pattern] = temp + get_syndrome(temp);/* add redundancy */
+ }
+
+ /*
+ * ---------------------------------------------------------------------
+ * Generate DECODING TABLE
+ *
+ * An entry to the decoding table is a syndrome and the resulting value
+ * is the most likely error pattern. First an error pattern is generated.
+ * Then its syndrome is calculated and used as a pointer to the table
+ * where the error pattern value is stored.
+ * ---------------------------------------------------------------------
+ *
+ * (1) Error patterns of WEIGHT 1 (SINGLE ERRORS)
+ */
+ decoding_table[0] = 0;
+ decoding_table[1] = 1;
+ temp = 1;
+ for (i=2; i<= 23; i++) {
+ temp *= 2;
+ decoding_table[get_syndrome(temp)] = temp;
+ }
+ /*
+ * (2) Error patterns of WEIGHT 2 (DOUBLE ERRORS)
+ */
+ a[1] = 1; a[2] = 2;
+ temp = arr2int(a,2);
+ decoding_table[get_syndrome(temp)] = temp;
+ for (i=1; i<253; i++) {
+ nextcomb(23,2,a);
+ temp = arr2int(a,2);
+ decoding_table[get_syndrome(temp)] = temp;
+ }
+ /*
+ * (3) Error patterns of WEIGHT 3 (TRIPLE ERRORS)
+ */
+ a[1] = 1; a[2] = 2; a[3] = 3;
+ temp = arr2int(a,3);
+ decoding_table[get_syndrome(temp)] = temp;
+ for (i=1; i<1771; i++) {
+ nextcomb(23,3,a);
+ temp = arr2int(a,3);
+ decoding_table[get_syndrome(temp)] = temp;
+ }
+
+ /* ---------------------------------------------------------------------
+ * Generate DATA
+ * ---------------------------------------------------------------------
+ */
+
+ /* Test all combinations of data and 1,2 or 3 errors */
+
+ tests = 0;
+ error_pattern = 1;
+ for (i=0; i< 23; i++) {
+ //printf("error_pattern: 0x%x\n", error_pattern);
+ tests += golay23_test(error_pattern);
+ error_pattern *= 2;
+ }
+ printf("%d 1 bit error tests performed OK!\n", tests);
+
+ tests = 0;
+ a[1] = 1; a[2] = 2;
+ error_pattern = arr2int(a,2);
+ tests += golay23_test(error_pattern);
+ for (i=1; i<253; i++) {
+ nextcomb(23,2,a);
+ error_pattern = arr2int(a,2);
+ //printf("error_pattern: 0x%x\n", error_pattern);
+ tests += golay23_test(error_pattern);
+ }
+ printf("%d 2 bit error tests performed OK!\n", tests);
+
+ tests = 0;
+ a[1] = 1; a[2] = 2; a[3] = 3;
+ error_pattern = arr2int(a,3);
+ tests += golay23_test(error_pattern);
+ for (i=1; i<1771; i++) {
+ nextcomb(23,3,a);
+ error_pattern = arr2int(a,3);
+ //printf("error_pattern: 0x%x\n", error_pattern);
+ tests += golay23_test(error_pattern);
+ }
+ printf("%d 3 bit error tests performed OK!\n", tests);
+
+ return 0;
+}
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