patch-2.1.66 linux/drivers/char/ftape/lowlevel/ftape-ecc.c
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- Lines: 854
- Date:
Tue Nov 25 14:45:27 1997
- Orig file:
v2.1.65/linux/drivers/char/ftape/lowlevel/ftape-ecc.c
- Orig date:
Wed Dec 31 16:00:00 1969
diff -u --recursive --new-file v2.1.65/linux/drivers/char/ftape/lowlevel/ftape-ecc.c linux/drivers/char/ftape/lowlevel/ftape-ecc.c
@@ -0,0 +1,853 @@
+/*
+ *
+ * Copyright (c) 1993 Ning and David Mosberger.
+
+ This is based on code originally written by Bas Laarhoven (bas@vimec.nl)
+ and David L. Brown, Jr., and incorporates improvements suggested by
+ Kai Harrekilde-Petersen.
+
+ This program is free software; you can redistribute it and/or
+ modify it under the terms of the GNU General Public License as
+ published by the Free Software Foundation; either version 2, or (at
+ your option) any later version.
+
+ This program is distributed in the hope that it will be useful, but
+ WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program; see the file COPYING. If not, write to
+ the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139,
+ USA.
+
+ *
+ * $Source: /homes/cvs/ftape-stacked/ftape/lowlevel/ftape-ecc.c,v $
+ * $Revision: 1.3 $
+ * $Date: 1997/10/05 19:18:10 $
+ *
+ * This file contains the Reed-Solomon error correction code
+ * for the QIC-40/80 floppy-tape driver for Linux.
+ */
+
+#include <linux/ftape.h>
+
+#include "../lowlevel/ftape-tracing.h"
+#include "../lowlevel/ftape-ecc.h"
+
+/* Machines that are big-endian should define macro BIG_ENDIAN.
+ * Unfortunately, there doesn't appear to be a standard include file
+ * that works for all OSs.
+ */
+
+#if defined(__sparc__) || defined(__hppa)
+#define BIG_ENDIAN
+#endif /* __sparc__ || __hppa */
+
+#if defined(__mips__)
+#error Find a smart way to determine the Endianness of the MIPS CPU
+#endif
+
+/* Notice: to minimize the potential for confusion, we use r to
+ * denote the independent variable of the polynomials in the
+ * Galois Field GF(2^8). We reserve x for polynomials that
+ * that have coefficients in GF(2^8).
+ *
+ * The Galois Field in which coefficient arithmetic is performed are
+ * the polynomials over Z_2 (i.e., 0 and 1) modulo the irreducible
+ * polynomial f(r), where f(r)=r^8 + r^7 + r^2 + r + 1. A polynomial
+ * is represented as a byte with the MSB as the coefficient of r^7 and
+ * the LSB as the coefficient of r^0. For example, the binary
+ * representation of f(x) is 0x187 (of course, this doesn't fit into 8
+ * bits). In this field, the polynomial r is a primitive element.
+ * That is, r^i with i in 0,...,255 enumerates all elements in the
+ * field.
+ *
+ * The generator polynomial for the QIC-80 ECC is
+ *
+ * g(x) = x^3 + r^105*x^2 + r^105*x + 1
+ *
+ * which can be factored into:
+ *
+ * g(x) = (x-r^-1)(x-r^0)(x-r^1)
+ *
+ * the byte representation of the coefficients are:
+ *
+ * r^105 = 0xc0
+ * r^-1 = 0xc3
+ * r^0 = 0x01
+ * r^1 = 0x02
+ *
+ * Notice that r^-1 = r^254 as exponent arithmetic is performed
+ * modulo 2^8-1 = 255.
+ *
+ * For more information on Galois Fields and Reed-Solomon codes, refer
+ * to any good book. I found _An Introduction to Error Correcting
+ * Codes with Applications_ by S. A. Vanstone and P. C. van Oorschot
+ * to be a good introduction into the former. _CODING THEORY: The
+ * Essentials_ I found very useful for its concise description of
+ * Reed-Solomon encoding/decoding.
+ *
+ */
+
+typedef __u8 Matrix[3][3];
+
+/*
+ * gfpow[] is defined such that gfpow[i] returns r^i if
+ * i is in the range [0..255].
+ */
+static const __u8 gfpow[] =
+{
+ 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80,
+ 0x87, 0x89, 0x95, 0xad, 0xdd, 0x3d, 0x7a, 0xf4,
+ 0x6f, 0xde, 0x3b, 0x76, 0xec, 0x5f, 0xbe, 0xfb,
+ 0x71, 0xe2, 0x43, 0x86, 0x8b, 0x91, 0xa5, 0xcd,
+ 0x1d, 0x3a, 0x74, 0xe8, 0x57, 0xae, 0xdb, 0x31,
+ 0x62, 0xc4, 0x0f, 0x1e, 0x3c, 0x78, 0xf0, 0x67,
+ 0xce, 0x1b, 0x36, 0x6c, 0xd8, 0x37, 0x6e, 0xdc,
+ 0x3f, 0x7e, 0xfc, 0x7f, 0xfe, 0x7b, 0xf6, 0x6b,
+ 0xd6, 0x2b, 0x56, 0xac, 0xdf, 0x39, 0x72, 0xe4,
+ 0x4f, 0x9e, 0xbb, 0xf1, 0x65, 0xca, 0x13, 0x26,
+ 0x4c, 0x98, 0xb7, 0xe9, 0x55, 0xaa, 0xd3, 0x21,
+ 0x42, 0x84, 0x8f, 0x99, 0xb5, 0xed, 0x5d, 0xba,
+ 0xf3, 0x61, 0xc2, 0x03, 0x06, 0x0c, 0x18, 0x30,
+ 0x60, 0xc0, 0x07, 0x0e, 0x1c, 0x38, 0x70, 0xe0,
+ 0x47, 0x8e, 0x9b, 0xb1, 0xe5, 0x4d, 0x9a, 0xb3,
+ 0xe1, 0x45, 0x8a, 0x93, 0xa1, 0xc5, 0x0d, 0x1a,
+ 0x34, 0x68, 0xd0, 0x27, 0x4e, 0x9c, 0xbf, 0xf9,
+ 0x75, 0xea, 0x53, 0xa6, 0xcb, 0x11, 0x22, 0x44,
+ 0x88, 0x97, 0xa9, 0xd5, 0x2d, 0x5a, 0xb4, 0xef,
+ 0x59, 0xb2, 0xe3, 0x41, 0x82, 0x83, 0x81, 0x85,
+ 0x8d, 0x9d, 0xbd, 0xfd, 0x7d, 0xfa, 0x73, 0xe6,
+ 0x4b, 0x96, 0xab, 0xd1, 0x25, 0x4a, 0x94, 0xaf,
+ 0xd9, 0x35, 0x6a, 0xd4, 0x2f, 0x5e, 0xbc, 0xff,
+ 0x79, 0xf2, 0x63, 0xc6, 0x0b, 0x16, 0x2c, 0x58,
+ 0xb0, 0xe7, 0x49, 0x92, 0xa3, 0xc1, 0x05, 0x0a,
+ 0x14, 0x28, 0x50, 0xa0, 0xc7, 0x09, 0x12, 0x24,
+ 0x48, 0x90, 0xa7, 0xc9, 0x15, 0x2a, 0x54, 0xa8,
+ 0xd7, 0x29, 0x52, 0xa4, 0xcf, 0x19, 0x32, 0x64,
+ 0xc8, 0x17, 0x2e, 0x5c, 0xb8, 0xf7, 0x69, 0xd2,
+ 0x23, 0x46, 0x8c, 0x9f, 0xb9, 0xf5, 0x6d, 0xda,
+ 0x33, 0x66, 0xcc, 0x1f, 0x3e, 0x7c, 0xf8, 0x77,
+ 0xee, 0x5b, 0xb6, 0xeb, 0x51, 0xa2, 0xc3, 0x01
+};
+
+/*
+ * This is a log table. That is, gflog[r^i] returns i (modulo f(r)).
+ * gflog[0] is undefined and the first element is therefore not valid.
+ */
+static const __u8 gflog[256] =
+{
+ 0xff, 0x00, 0x01, 0x63, 0x02, 0xc6, 0x64, 0x6a,
+ 0x03, 0xcd, 0xc7, 0xbc, 0x65, 0x7e, 0x6b, 0x2a,
+ 0x04, 0x8d, 0xce, 0x4e, 0xc8, 0xd4, 0xbd, 0xe1,
+ 0x66, 0xdd, 0x7f, 0x31, 0x6c, 0x20, 0x2b, 0xf3,
+ 0x05, 0x57, 0x8e, 0xe8, 0xcf, 0xac, 0x4f, 0x83,
+ 0xc9, 0xd9, 0xd5, 0x41, 0xbe, 0x94, 0xe2, 0xb4,
+ 0x67, 0x27, 0xde, 0xf0, 0x80, 0xb1, 0x32, 0x35,
+ 0x6d, 0x45, 0x21, 0x12, 0x2c, 0x0d, 0xf4, 0x38,
+ 0x06, 0x9b, 0x58, 0x1a, 0x8f, 0x79, 0xe9, 0x70,
+ 0xd0, 0xc2, 0xad, 0xa8, 0x50, 0x75, 0x84, 0x48,
+ 0xca, 0xfc, 0xda, 0x8a, 0xd6, 0x54, 0x42, 0x24,
+ 0xbf, 0x98, 0x95, 0xf9, 0xe3, 0x5e, 0xb5, 0x15,
+ 0x68, 0x61, 0x28, 0xba, 0xdf, 0x4c, 0xf1, 0x2f,
+ 0x81, 0xe6, 0xb2, 0x3f, 0x33, 0xee, 0x36, 0x10,
+ 0x6e, 0x18, 0x46, 0xa6, 0x22, 0x88, 0x13, 0xf7,
+ 0x2d, 0xb8, 0x0e, 0x3d, 0xf5, 0xa4, 0x39, 0x3b,
+ 0x07, 0x9e, 0x9c, 0x9d, 0x59, 0x9f, 0x1b, 0x08,
+ 0x90, 0x09, 0x7a, 0x1c, 0xea, 0xa0, 0x71, 0x5a,
+ 0xd1, 0x1d, 0xc3, 0x7b, 0xae, 0x0a, 0xa9, 0x91,
+ 0x51, 0x5b, 0x76, 0x72, 0x85, 0xa1, 0x49, 0xeb,
+ 0xcb, 0x7c, 0xfd, 0xc4, 0xdb, 0x1e, 0x8b, 0xd2,
+ 0xd7, 0x92, 0x55, 0xaa, 0x43, 0x0b, 0x25, 0xaf,
+ 0xc0, 0x73, 0x99, 0x77, 0x96, 0x5c, 0xfa, 0x52,
+ 0xe4, 0xec, 0x5f, 0x4a, 0xb6, 0xa2, 0x16, 0x86,
+ 0x69, 0xc5, 0x62, 0xfe, 0x29, 0x7d, 0xbb, 0xcc,
+ 0xe0, 0xd3, 0x4d, 0x8c, 0xf2, 0x1f, 0x30, 0xdc,
+ 0x82, 0xab, 0xe7, 0x56, 0xb3, 0x93, 0x40, 0xd8,
+ 0x34, 0xb0, 0xef, 0x26, 0x37, 0x0c, 0x11, 0x44,
+ 0x6f, 0x78, 0x19, 0x9a, 0x47, 0x74, 0xa7, 0xc1,
+ 0x23, 0x53, 0x89, 0xfb, 0x14, 0x5d, 0xf8, 0x97,
+ 0x2e, 0x4b, 0xb9, 0x60, 0x0f, 0xed, 0x3e, 0xe5,
+ 0xf6, 0x87, 0xa5, 0x17, 0x3a, 0xa3, 0x3c, 0xb7
+};
+
+/* This is a multiplication table for the factor 0xc0 (i.e., r^105 (mod f(r)).
+ * gfmul_c0[f] returns r^105 * f(r) (modulo f(r)).
+ */
+static const __u8 gfmul_c0[256] =
+{
+ 0x00, 0xc0, 0x07, 0xc7, 0x0e, 0xce, 0x09, 0xc9,
+ 0x1c, 0xdc, 0x1b, 0xdb, 0x12, 0xd2, 0x15, 0xd5,
+ 0x38, 0xf8, 0x3f, 0xff, 0x36, 0xf6, 0x31, 0xf1,
+ 0x24, 0xe4, 0x23, 0xe3, 0x2a, 0xea, 0x2d, 0xed,
+ 0x70, 0xb0, 0x77, 0xb7, 0x7e, 0xbe, 0x79, 0xb9,
+ 0x6c, 0xac, 0x6b, 0xab, 0x62, 0xa2, 0x65, 0xa5,
+ 0x48, 0x88, 0x4f, 0x8f, 0x46, 0x86, 0x41, 0x81,
+ 0x54, 0x94, 0x53, 0x93, 0x5a, 0x9a, 0x5d, 0x9d,
+ 0xe0, 0x20, 0xe7, 0x27, 0xee, 0x2e, 0xe9, 0x29,
+ 0xfc, 0x3c, 0xfb, 0x3b, 0xf2, 0x32, 0xf5, 0x35,
+ 0xd8, 0x18, 0xdf, 0x1f, 0xd6, 0x16, 0xd1, 0x11,
+ 0xc4, 0x04, 0xc3, 0x03, 0xca, 0x0a, 0xcd, 0x0d,
+ 0x90, 0x50, 0x97, 0x57, 0x9e, 0x5e, 0x99, 0x59,
+ 0x8c, 0x4c, 0x8b, 0x4b, 0x82, 0x42, 0x85, 0x45,
+ 0xa8, 0x68, 0xaf, 0x6f, 0xa6, 0x66, 0xa1, 0x61,
+ 0xb4, 0x74, 0xb3, 0x73, 0xba, 0x7a, 0xbd, 0x7d,
+ 0x47, 0x87, 0x40, 0x80, 0x49, 0x89, 0x4e, 0x8e,
+ 0x5b, 0x9b, 0x5c, 0x9c, 0x55, 0x95, 0x52, 0x92,
+ 0x7f, 0xbf, 0x78, 0xb8, 0x71, 0xb1, 0x76, 0xb6,
+ 0x63, 0xa3, 0x64, 0xa4, 0x6d, 0xad, 0x6a, 0xaa,
+ 0x37, 0xf7, 0x30, 0xf0, 0x39, 0xf9, 0x3e, 0xfe,
+ 0x2b, 0xeb, 0x2c, 0xec, 0x25, 0xe5, 0x22, 0xe2,
+ 0x0f, 0xcf, 0x08, 0xc8, 0x01, 0xc1, 0x06, 0xc6,
+ 0x13, 0xd3, 0x14, 0xd4, 0x1d, 0xdd, 0x1a, 0xda,
+ 0xa7, 0x67, 0xa0, 0x60, 0xa9, 0x69, 0xae, 0x6e,
+ 0xbb, 0x7b, 0xbc, 0x7c, 0xb5, 0x75, 0xb2, 0x72,
+ 0x9f, 0x5f, 0x98, 0x58, 0x91, 0x51, 0x96, 0x56,
+ 0x83, 0x43, 0x84, 0x44, 0x8d, 0x4d, 0x8a, 0x4a,
+ 0xd7, 0x17, 0xd0, 0x10, 0xd9, 0x19, 0xde, 0x1e,
+ 0xcb, 0x0b, 0xcc, 0x0c, 0xc5, 0x05, 0xc2, 0x02,
+ 0xef, 0x2f, 0xe8, 0x28, 0xe1, 0x21, 0xe6, 0x26,
+ 0xf3, 0x33, 0xf4, 0x34, 0xfd, 0x3d, 0xfa, 0x3a
+};
+
+
+/* Returns V modulo 255 provided V is in the range -255,-254,...,509.
+ */
+static inline __u8 mod255(int v)
+{
+ if (v > 0) {
+ if (v < 255) {
+ return v;
+ } else {
+ return v - 255;
+ }
+ } else {
+ return v + 255;
+ }
+}
+
+
+/* Add two numbers in the field. Addition in this field is equivalent
+ * to a bit-wise exclusive OR operation---subtraction is therefore
+ * identical to addition.
+ */
+static inline __u8 gfadd(__u8 a, __u8 b)
+{
+ return a ^ b;
+}
+
+
+/* Add two vectors of numbers in the field. Each byte in A and B gets
+ * added individually.
+ */
+static inline unsigned long gfadd_long(unsigned long a, unsigned long b)
+{
+ return a ^ b;
+}
+
+
+/* Multiply two numbers in the field:
+ */
+static inline __u8 gfmul(__u8 a, __u8 b)
+{
+ if (a && b) {
+ return gfpow[mod255(gflog[a] + gflog[b])];
+ } else {
+ return 0;
+ }
+}
+
+
+/* Just like gfmul, except we have already looked up the log of the
+ * second number.
+ */
+static inline __u8 gfmul_exp(__u8 a, int b)
+{
+ if (a) {
+ return gfpow[mod255(gflog[a] + b)];
+ } else {
+ return 0;
+ }
+}
+
+
+/* Just like gfmul_exp, except that A is a vector of numbers. That
+ * is, each byte in A gets multiplied by gfpow[mod255(B)].
+ */
+static inline unsigned long gfmul_exp_long(unsigned long a, int b)
+{
+ __u8 t;
+
+ if (sizeof(long) == 4) {
+ return (
+ ((t = (__u32)a >> 24 & 0xff) ?
+ (((__u32) gfpow[mod255(gflog[t] + b)]) << 24) : 0) |
+ ((t = (__u32)a >> 16 & 0xff) ?
+ (((__u32) gfpow[mod255(gflog[t] + b)]) << 16) : 0) |
+ ((t = (__u32)a >> 8 & 0xff) ?
+ (((__u32) gfpow[mod255(gflog[t] + b)]) << 8) : 0) |
+ ((t = (__u32)a >> 0 & 0xff) ?
+ (((__u32) gfpow[mod255(gflog[t] + b)]) << 0) : 0));
+ } else if (sizeof(long) == 8) {
+ return (
+ ((t = (__u64)a >> 56 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 56) : 0) |
+ ((t = (__u64)a >> 48 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 48) : 0) |
+ ((t = (__u64)a >> 40 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 40) : 0) |
+ ((t = (__u64)a >> 32 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 32) : 0) |
+ ((t = (__u64)a >> 24 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 24) : 0) |
+ ((t = (__u64)a >> 16 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 16) : 0) |
+ ((t = (__u64)a >> 8 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 8) : 0) |
+ ((t = (__u64)a >> 0 & 0xff) ?
+ (((__u64) gfpow[mod255(gflog[t] + b)]) << 0) : 0));
+ } else {
+ TRACE_FUN(ft_t_any);
+ TRACE_ABORT(-1, ft_t_err, "Error: size of long is %d bytes",
+ (int)sizeof(long));
+ }
+}
+
+
+/* Divide two numbers in the field. Returns a/b (modulo f(x)).
+ */
+static inline __u8 gfdiv(__u8 a, __u8 b)
+{
+ if (!b) {
+ TRACE_FUN(ft_t_any);
+ TRACE_ABORT(0xff, ft_t_bug, "Error: division by zero");
+ } else if (a == 0) {
+ return 0;
+ } else {
+ return gfpow[mod255(gflog[a] - gflog[b])];
+ }
+}
+
+
+/* The following functions return the inverse of the matrix of the
+ * linear system that needs to be solved to determine the error
+ * magnitudes. The first deals with matrices of rank 3, while the
+ * second deals with matrices of rank 2. The error indices are passed
+ * in arguments L0,..,L2 (0=first sector, 31=last sector). The error
+ * indices must be sorted in ascending order, i.e., L0<L1<L2.
+ *
+ * The linear system that needs to be solved for the error magnitudes
+ * is A * b = s, where s is the known vector of syndromes, b is the
+ * vector of error magnitudes and A in the ORDER=3 case:
+ *
+ * A_3 = {{1/r^L[0], 1/r^L[1], 1/r^L[2]},
+ * { 1, 1, 1},
+ * { r^L[0], r^L[1], r^L[2]}}
+ */
+static inline int gfinv3(__u8 l0,
+ __u8 l1,
+ __u8 l2,
+ Matrix Ainv)
+{
+ __u8 det;
+ __u8 t20, t10, t21, t12, t01, t02;
+ int log_det;
+
+ /* compute some intermediate results: */
+ t20 = gfpow[l2 - l0]; /* t20 = r^l2/r^l0 */
+ t10 = gfpow[l1 - l0]; /* t10 = r^l1/r^l0 */
+ t21 = gfpow[l2 - l1]; /* t21 = r^l2/r^l1 */
+ t12 = gfpow[l1 - l2 + 255]; /* t12 = r^l1/r^l2 */
+ t01 = gfpow[l0 - l1 + 255]; /* t01 = r^l0/r^l1 */
+ t02 = gfpow[l0 - l2 + 255]; /* t02 = r^l0/r^l2 */
+ /* Calculate the determinant of matrix A_3^-1 (sometimes
+ * called the Vandermonde determinant):
+ */
+ det = gfadd(t20, gfadd(t10, gfadd(t21, gfadd(t12, gfadd(t01, t02)))));
+ if (!det) {
+ TRACE_FUN(ft_t_any);
+ TRACE_ABORT(0, ft_t_err,
+ "Inversion failed (3 CRC errors, >0 CRC failures)");
+ }
+ log_det = 255 - gflog[det];
+
+ /* Now, calculate all of the coefficients:
+ */
+ Ainv[0][0]= gfmul_exp(gfadd(gfpow[l1], gfpow[l2]), log_det);
+ Ainv[0][1]= gfmul_exp(gfadd(t21, t12), log_det);
+ Ainv[0][2]= gfmul_exp(gfadd(gfpow[255 - l1], gfpow[255 - l2]),log_det);
+
+ Ainv[1][0]= gfmul_exp(gfadd(gfpow[l0], gfpow[l2]), log_det);
+ Ainv[1][1]= gfmul_exp(gfadd(t20, t02), log_det);
+ Ainv[1][2]= gfmul_exp(gfadd(gfpow[255 - l0], gfpow[255 - l2]),log_det);
+
+ Ainv[2][0]= gfmul_exp(gfadd(gfpow[l0], gfpow[l1]), log_det);
+ Ainv[2][1]= gfmul_exp(gfadd(t10, t01), log_det);
+ Ainv[2][2]= gfmul_exp(gfadd(gfpow[255 - l0], gfpow[255 - l1]),log_det);
+
+ return 1;
+}
+
+
+static inline int gfinv2(__u8 l0, __u8 l1, Matrix Ainv)
+{
+ __u8 det;
+ __u8 t1, t2;
+ int log_det;
+
+ t1 = gfpow[255 - l0];
+ t2 = gfpow[255 - l1];
+ det = gfadd(t1, t2);
+ if (!det) {
+ TRACE_FUN(ft_t_any);
+ TRACE_ABORT(0, ft_t_err,
+ "Inversion failed (2 CRC errors, >0 CRC failures)");
+ }
+ log_det = 255 - gflog[det];
+
+ /* Now, calculate all of the coefficients:
+ */
+ Ainv[0][0] = Ainv[1][0] = gfpow[log_det];
+
+ Ainv[0][1] = gfmul_exp(t2, log_det);
+ Ainv[1][1] = gfmul_exp(t1, log_det);
+
+ return 1;
+}
+
+
+/* Multiply matrix A by vector S and return result in vector B. M is
+ * assumed to be of order NxN, S and B of order Nx1.
+ */
+static inline void gfmat_mul(int n, Matrix A,
+ __u8 *s, __u8 *b)
+{
+ int i, j;
+ __u8 dot_prod;
+
+ for (i = 0; i < n; ++i) {
+ dot_prod = 0;
+ for (j = 0; j < n; ++j) {
+ dot_prod = gfadd(dot_prod, gfmul(A[i][j], s[j]));
+ }
+ b[i] = dot_prod;
+ }
+}
+
+
+
+/* The Reed Solomon ECC codes are computed over the N-th byte of each
+ * block, where N=SECTOR_SIZE. There are up to 29 blocks of data, and
+ * 3 blocks of ECC. The blocks are stored contiguously in memory. A
+ * segment, consequently, is assumed to have at least 4 blocks: one or
+ * more data blocks plus three ECC blocks.
+ *
+ * Notice: In QIC-80 speak, a CRC error is a sector with an incorrect
+ * CRC. A CRC failure is a sector with incorrect data, but
+ * a valid CRC. In the error control literature, the former
+ * is usually called "erasure", the latter "error."
+ */
+/* Compute the parity bytes for C columns of data, where C is the
+ * number of bytes that fit into a long integer. We use a linear
+ * feed-back register to do this. The parity bytes P[0], P[STRIDE],
+ * P[2*STRIDE] are computed such that:
+ *
+ * x^k * p(x) + m(x) = 0 (modulo g(x))
+ *
+ * where k = NBLOCKS,
+ * p(x) = P[0] + P[STRIDE]*x + P[2*STRIDE]*x^2, and
+ * m(x) = sum_{i=0}^k m_i*x^i.
+ * m_i = DATA[i*SECTOR_SIZE]
+ */
+static inline void set_parity(unsigned long *data,
+ int nblocks,
+ unsigned long *p,
+ int stride)
+{
+ unsigned long p0, p1, p2, t1, t2, *end;
+
+ end = data + nblocks * (FT_SECTOR_SIZE / sizeof(long));
+ p0 = p1 = p2 = 0;
+ while (data < end) {
+ /* The new parity bytes p0_i, p1_i, p2_i are computed
+ * from the old values p0_{i-1}, p1_{i-1}, p2_{i-1}
+ * recursively as:
+ *
+ * p0_i = p1_{i-1} + r^105 * (m_{i-1} - p0_{i-1})
+ * p1_i = p2_{i-1} + r^105 * (m_{i-1} - p0_{i-1})
+ * p2_i = (m_{i-1} - p0_{i-1})
+ *
+ * With the initial condition: p0_0 = p1_0 = p2_0 = 0.
+ */
+ t1 = gfadd_long(*data, p0);
+ /*
+ * Multiply each byte in t1 by 0xc0:
+ */
+ if (sizeof(long) == 4) {
+ t2= (((__u32) gfmul_c0[(__u32)t1 >> 24 & 0xff]) << 24 |
+ ((__u32) gfmul_c0[(__u32)t1 >> 16 & 0xff]) << 16 |
+ ((__u32) gfmul_c0[(__u32)t1 >> 8 & 0xff]) << 8 |
+ ((__u32) gfmul_c0[(__u32)t1 >> 0 & 0xff]) << 0);
+ } else if (sizeof(long) == 8) {
+ t2= (((__u64) gfmul_c0[(__u64)t1 >> 56 & 0xff]) << 56 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 48 & 0xff]) << 48 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 40 & 0xff]) << 40 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 32 & 0xff]) << 32 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 24 & 0xff]) << 24 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 16 & 0xff]) << 16 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 8 & 0xff]) << 8 |
+ ((__u64) gfmul_c0[(__u64)t1 >> 0 & 0xff]) << 0);
+ } else {
+ TRACE_FUN(ft_t_any);
+ TRACE(ft_t_err, "Error: long is of size %d",
+ (int) sizeof(long));
+ TRACE_EXIT;
+ }
+ p0 = gfadd_long(t2, p1);
+ p1 = gfadd_long(t2, p2);
+ p2 = t1;
+ data += FT_SECTOR_SIZE / sizeof(long);
+ }
+ *p = p0;
+ p += stride;
+ *p = p1;
+ p += stride;
+ *p = p2;
+ return;
+}
+
+
+/* Compute the 3 syndrome values. DATA should point to the first byte
+ * of the column for which the syndromes are desired. The syndromes
+ * are computed over the first NBLOCKS of rows. The three bytes will
+ * be placed in S[0], S[1], and S[2].
+ *
+ * S[i] is the value of the "message" polynomial m(x) evaluated at the
+ * i-th root of the generator polynomial g(x).
+ *
+ * As g(x)=(x-r^-1)(x-1)(x-r^1) we evaluate the message polynomial at
+ * x=r^-1 to get S[0], at x=r^0=1 to get S[1], and at x=r to get S[2].
+ * This could be done directly and efficiently via the Horner scheme.
+ * However, it would require multiplication tables for the factors
+ * r^-1 (0xc3) and r (0x02). The following scheme does not require
+ * any multiplication tables beyond what's needed for set_parity()
+ * anyway and is slightly faster if there are no errors and slightly
+ * slower if there are errors. The latter is hopefully the infrequent
+ * case.
+ *
+ * To understand the alternative algorithm, notice that set_parity(m,
+ * k, p) computes parity bytes such that:
+ *
+ * x^k * p(x) = m(x) (modulo g(x)).
+ *
+ * That is, to evaluate m(r^m), where r^m is a root of g(x), we can
+ * simply evaluate (r^m)^k*p(r^m). Also, notice that p is 0 if and
+ * only if s is zero. That is, if all parity bytes are 0, we know
+ * there is no error in the data and consequently there is no need to
+ * compute s(x) at all! In all other cases, we compute s(x) from p(x)
+ * by evaluating (r^m)^k*p(r^m) for m=-1, m=0, and m=1. The p(x)
+ * polynomial is evaluated via the Horner scheme.
+ */
+static int compute_syndromes(unsigned long *data, int nblocks, unsigned long *s)
+{
+ unsigned long p[3];
+
+ set_parity(data, nblocks, p, 1);
+ if (p[0] | p[1] | p[2]) {
+ /* Some of the checked columns do not have a zero
+ * syndrome. For simplicity, we compute the syndromes
+ * for all columns that we have computed the
+ * remainders for.
+ */
+ s[0] = gfmul_exp_long(
+ gfadd_long(p[0],
+ gfmul_exp_long(
+ gfadd_long(p[1],
+ gfmul_exp_long(p[2], -1)),
+ -1)),
+ -nblocks);
+ s[1] = gfadd_long(gfadd_long(p[2], p[1]), p[0]);
+ s[2] = gfmul_exp_long(
+ gfadd_long(p[0],
+ gfmul_exp_long(
+ gfadd_long(p[1],
+ gfmul_exp_long(p[2], 1)),
+ 1)),
+ nblocks);
+ return 0;
+ } else {
+ return 1;
+ }
+}
+
+
+/* Correct the block in the column pointed to by DATA. There are NBAD
+ * CRC errors and their indices are in BAD_LOC[0], up to
+ * BAD_LOC[NBAD-1]. If NBAD>1, Ainv holds the inverse of the matrix
+ * of the linear system that needs to be solved to determine the error
+ * magnitudes. S[0], S[1], and S[2] are the syndrome values. If row
+ * j gets corrected, then bit j will be set in CORRECTION_MAP.
+ */
+static inline int correct_block(__u8 *data, int nblocks,
+ int nbad, int *bad_loc, Matrix Ainv,
+ __u8 *s,
+ SectorMap * correction_map)
+{
+ int ncorrected = 0;
+ int i;
+ __u8 t1, t2;
+ __u8 c0, c1, c2; /* check bytes */
+ __u8 error_mag[3], log_error_mag;
+ __u8 *dp, l, e;
+ TRACE_FUN(ft_t_any);
+
+ switch (nbad) {
+ case 0:
+ /* might have a CRC failure: */
+ if (s[0] == 0) {
+ /* more than one error */
+ TRACE_ABORT(-1, ft_t_err,
+ "ECC failed (0 CRC errors, >1 CRC failures)");
+ }
+ t1 = gfdiv(s[1], s[0]);
+ if ((bad_loc[nbad++] = gflog[t1]) >= nblocks) {
+ TRACE(ft_t_err,
+ "ECC failed (0 CRC errors, >1 CRC failures)");
+ TRACE_ABORT(-1, ft_t_err,
+ "attempt to correct data at %d", bad_loc[0]);
+ }
+ error_mag[0] = s[1];
+ break;
+ case 1:
+ t1 = gfadd(gfmul_exp(s[1], bad_loc[0]), s[2]);
+ t2 = gfadd(gfmul_exp(s[0], bad_loc[0]), s[1]);
+ if (t1 == 0 && t2 == 0) {
+ /* one erasure, no error: */
+ Ainv[0][0] = gfpow[bad_loc[0]];
+ } else if (t1 == 0 || t2 == 0) {
+ /* one erasure and more than one error: */
+ TRACE_ABORT(-1, ft_t_err,
+ "ECC failed (1 erasure, >1 error)");
+ } else {
+ /* one erasure, one error: */
+ if ((bad_loc[nbad++] = gflog[gfdiv(t1, t2)])
+ >= nblocks) {
+ TRACE(ft_t_err, "ECC failed "
+ "(1 CRC errors, >1 CRC failures)");
+ TRACE_ABORT(-1, ft_t_err,
+ "attempt to correct data at %d",
+ bad_loc[1]);
+ }
+ if (!gfinv2(bad_loc[0], bad_loc[1], Ainv)) {
+ /* inversion failed---must have more
+ * than one error
+ */
+ TRACE_EXIT -1;
+ }
+ }
+ /* FALL THROUGH TO ERROR MAGNITUDE COMPUTATION:
+ */
+ case 2:
+ case 3:
+ /* compute error magnitudes: */
+ gfmat_mul(nbad, Ainv, s, error_mag);
+ break;
+
+ default:
+ TRACE_ABORT(-1, ft_t_err,
+ "Internal Error: number of CRC errors > 3");
+ }
+
+ /* Perform correction by adding ERROR_MAG[i] to the byte at
+ * offset BAD_LOC[i]. Also add the value of the computed
+ * error polynomial to the syndrome values. If the correction
+ * was successful, the resulting check bytes should be zero
+ * (i.e., the corrected data is a valid code word).
+ */
+ c0 = s[0];
+ c1 = s[1];
+ c2 = s[2];
+ for (i = 0; i < nbad; ++i) {
+ e = error_mag[i];
+ if (e) {
+ /* correct the byte at offset L by magnitude E: */
+ l = bad_loc[i];
+ dp = &data[l * FT_SECTOR_SIZE];
+ *dp = gfadd(*dp, e);
+ *correction_map |= 1 << l;
+ ++ncorrected;
+
+ log_error_mag = gflog[e];
+ c0 = gfadd(c0, gfpow[mod255(log_error_mag - l)]);
+ c1 = gfadd(c1, e);
+ c2 = gfadd(c2, gfpow[mod255(log_error_mag + l)]);
+ }
+ }
+ if (c0 || c1 || c2) {
+ TRACE_ABORT(-1, ft_t_err,
+ "ECC self-check failed, too many errors");
+ }
+ TRACE_EXIT ncorrected;
+}
+
+
+#if defined(ECC_SANITY_CHECK) || defined(ECC_PARANOID)
+
+/* Perform a sanity check on the computed parity bytes:
+ */
+static int sanity_check(unsigned long *data, int nblocks)
+{
+ TRACE_FUN(ft_t_any);
+ unsigned long s[3];
+
+ if (!compute_syndromes(data, nblocks, s)) {
+ TRACE_ABORT(0, ft_bug,
+ "Internal Error: syndrome self-check failed");
+ }
+ TRACE_EXIT 1;
+}
+
+#endif /* defined(ECC_SANITY_CHECK) || defined(ECC_PARANOID) */
+
+/* Compute the parity for an entire segment of data.
+ */
+int ftape_ecc_set_segment_parity(struct memory_segment *mseg)
+{
+ int i;
+ __u8 *parity_bytes;
+
+ parity_bytes = &mseg->data[(mseg->blocks - 3) * FT_SECTOR_SIZE];
+ for (i = 0; i < FT_SECTOR_SIZE; i += sizeof(long)) {
+ set_parity((unsigned long *) &mseg->data[i], mseg->blocks - 3,
+ (unsigned long *) &parity_bytes[i],
+ FT_SECTOR_SIZE / sizeof(long));
+#ifdef ECC_PARANOID
+ if (!sanity_check((unsigned long *) &mseg->data[i],
+ mseg->blocks)) {
+ return -1;
+ }
+#endif /* ECC_PARANOID */
+ }
+ return 0;
+}
+
+
+/* Checks and corrects (if possible) the segment MSEG. Returns one of
+ * ECC_OK, ECC_CORRECTED, and ECC_FAILED.
+ */
+int ftape_ecc_correct_data(struct memory_segment *mseg)
+{
+ int col, i, result;
+ int ncorrected = 0;
+ int nerasures = 0; /* # of erasures (CRC errors) */
+ int erasure_loc[3]; /* erasure locations */
+ unsigned long ss[3];
+ __u8 s[3];
+ Matrix Ainv;
+ TRACE_FUN(ft_t_flow);
+
+ mseg->corrected = 0;
+
+ /* find first column that has non-zero syndromes: */
+ for (col = 0; col < FT_SECTOR_SIZE; col += sizeof(long)) {
+ if (!compute_syndromes((unsigned long *) &mseg->data[col],
+ mseg->blocks, ss)) {
+ /* something is wrong---have to fix things */
+ break;
+ }
+ }
+ if (col >= FT_SECTOR_SIZE) {
+ /* all syndromes are ok, therefore nothing to correct */
+ TRACE_EXIT ECC_OK;
+ }
+ /* count the number of CRC errors if there were any: */
+ if (mseg->read_bad) {
+ for (i = 0; i < mseg->blocks; i++) {
+ if (BAD_CHECK(mseg->read_bad, i)) {
+ if (nerasures >= 3) {
+ /* this is too much for ECC */
+ TRACE_ABORT(ECC_FAILED, ft_t_err,
+ "ECC failed (>3 CRC errors)");
+ } /* if */
+ erasure_loc[nerasures++] = i;
+ }
+ }
+ }
+ /*
+ * If there are at least 2 CRC errors, determine inverse of matrix
+ * of linear system to be solved:
+ */
+ switch (nerasures) {
+ case 2:
+ if (!gfinv2(erasure_loc[0], erasure_loc[1], Ainv)) {
+ TRACE_EXIT ECC_FAILED;
+ }
+ break;
+ case 3:
+ if (!gfinv3(erasure_loc[0], erasure_loc[1],
+ erasure_loc[2], Ainv)) {
+ TRACE_EXIT ECC_FAILED;
+ }
+ break;
+ default:
+ /* this is not an error condition... */
+ break;
+ }
+
+ do {
+ for (i = 0; i < sizeof(long); ++i) {
+ s[0] = ss[0];
+ s[1] = ss[1];
+ s[2] = ss[2];
+ if (s[0] | s[1] | s[2]) {
+#ifdef BIG_ENDIAN
+ result = correct_block(
+ &mseg->data[col + sizeof(long) - 1 - i],
+ mseg->blocks,
+ nerasures,
+ erasure_loc,
+ Ainv,
+ s,
+ &mseg->corrected);
+#else
+ result = correct_block(&mseg->data[col + i],
+ mseg->blocks,
+ nerasures,
+ erasure_loc,
+ Ainv,
+ s,
+ &mseg->corrected);
+#endif
+ if (result < 0) {
+ TRACE_EXIT ECC_FAILED;
+ }
+ ncorrected += result;
+ }
+ ss[0] >>= 8;
+ ss[1] >>= 8;
+ ss[2] >>= 8;
+ }
+
+#ifdef ECC_SANITY_CHECK
+ if (!sanity_check((unsigned long *) &mseg->data[col],
+ mseg->blocks)) {
+ TRACE_EXIT ECC_FAILED;
+ }
+#endif /* ECC_SANITY_CHECK */
+
+ /* find next column with non-zero syndromes: */
+ while ((col += sizeof(long)) < FT_SECTOR_SIZE) {
+ if (!compute_syndromes((unsigned long *)
+ &mseg->data[col], mseg->blocks, ss)) {
+ /* something is wrong---have to fix things */
+ break;
+ }
+ }
+ } while (col < FT_SECTOR_SIZE);
+ if (ncorrected && nerasures == 0) {
+ TRACE(ft_t_warn, "block contained error not caught by CRC");
+ }
+ TRACE((ncorrected > 0) ? ft_t_noise : ft_t_any, "number of corrections: %d", ncorrected);
+ TRACE_EXIT ncorrected ? ECC_CORRECTED : ECC_OK;
+}
FUNET's LINUX-ADM group, linux-adm@nic.funet.fi
TCL-scripts by Sam Shen, slshen@lbl.gov