From 0cf544e3aa5468bc19cc7c45c128a1ef525e7506 Mon Sep 17 00:00:00 2001
From: Joe Taylor <joe@princeton.edu>
Date: Thu, 1 Feb 2024 08:52:46 -0500
Subject: [PATCH] Add modified versions of Karn RS routines.

---
 lib/superfox/decode_rs.c | 263 +++++++++++++++++++++++++++++++++++++++
 lib/superfox/encode_rs.c |  52 ++++++++
 lib/superfox/init_rs.c   | 126 +++++++++++++++++++
 lib/superfox/int_sf.h    |  56 +++++++++
 lib/superfox/rs_sf.c     |  57 +++++++++
 lib/superfox/rs_sf.h     |  35 ++++++
 6 files changed, 589 insertions(+)
 create mode 100644 lib/superfox/decode_rs.c
 create mode 100644 lib/superfox/encode_rs.c
 create mode 100644 lib/superfox/init_rs.c
 create mode 100644 lib/superfox/int_sf.h
 create mode 100644 lib/superfox/rs_sf.c
 create mode 100644 lib/superfox/rs_sf.h

diff --git a/lib/superfox/decode_rs.c b/lib/superfox/decode_rs.c
new file mode 100644
index 000000000..75e1978f3
--- /dev/null
+++ b/lib/superfox/decode_rs.c
@@ -0,0 +1,263 @@
+/* Reed-Solomon decoder
+ * Copyright 2002 Phil Karn, KA9Q
+ * May be used under the terms of the GNU General Public License (GPL)
+ */
+
+#ifdef DEBUG
+#include <stdio.h>
+#endif
+
+#include <string.h>
+
+#define NULL ((void *)0)
+#define	min(a,b)	((a) < (b) ? (a) : (b))
+
+#ifdef FIXED
+#include "fixed.h"
+#elif defined(BIGSYM)
+#include "int_sf.h"
+#else
+#include "char.h"
+#endif
+
+int DECODE_RS(
+#ifdef FIXED
+DTYPE *data, int *eras_pos, int no_eras,int pad){
+#else
+void *p,DTYPE *data, int *eras_pos, int no_eras){
+  struct rs *rs = (struct rs *)p;
+#endif
+  int deg_lambda, el, deg_omega;
+  int i, j, r,k;
+  DTYPE u,q,tmp,num1,num2,den,discr_r;
+  DTYPE lambda[NROOTS+1], s[NROOTS];	/* Err+Eras Locator poly
+					 * and syndrome poly */
+  DTYPE b[NROOTS+1], t[NROOTS+1], omega[NROOTS+1];
+  DTYPE root[NROOTS], reg[NROOTS+1], loc[NROOTS];
+  int syn_error, count;
+
+#ifdef FIXED
+  /* Check pad parameter for validity */
+  if(pad < 0 || pad >= NN)
+    return -1;
+#endif
+
+  /* form the syndromes; i.e., evaluate data(x) at roots of g(x) */
+  for(i=0;i<NROOTS;i++)
+    s[i] = data[0];
+
+  for(j=1;j<NN-PAD;j++){
+    for(i=0;i<NROOTS;i++){
+      if(s[i] == 0){
+	s[i] = data[j];
+      } else {
+	s[i] = data[j] ^ ALPHA_TO[MODNN(INDEX_OF[s[i]] + (FCR+i)*PRIM)];
+      }
+    }
+  }
+
+  /* Convert syndromes to index form, checking for nonzero condition */
+  syn_error = 0;
+  for(i=0;i<NROOTS;i++){
+    syn_error |= s[i];
+    s[i] = INDEX_OF[s[i]];
+  }
+
+  if (!syn_error) {
+    /* if syndrome is zero, data[] is a codeword and there are no
+     * errors to correct. So return data[] unmodified
+     */
+    count = 0;
+    goto finish;
+  }
+  memset(&lambda[1],0,NROOTS*sizeof(lambda[0]));
+  lambda[0] = 1;
+
+  if (no_eras > 0) {
+    /* Init lambda to be the erasure locator polynomial */
+    lambda[1] = ALPHA_TO[MODNN(PRIM*(NN-1-eras_pos[0]))];
+    for (i = 1; i < no_eras; i++) {
+      u = MODNN(PRIM*(NN-1-eras_pos[i]));
+      for (j = i+1; j > 0; j--) {
+	tmp = INDEX_OF[lambda[j - 1]];
+	if(tmp != A0)
+	  lambda[j] ^= ALPHA_TO[MODNN(u + tmp)];
+      }
+    }
+
+#if DEBUG >= 1
+    /* Test code that verifies the erasure locator polynomial just constructed
+       Needed only for decoder debugging. */
+    
+    /* find roots of the erasure location polynomial */
+    for(i=1;i<=no_eras;i++)
+      reg[i] = INDEX_OF[lambda[i]];
+
+    count = 0;
+    for (i = 1,k=IPRIM-1; i <= NN; i++,k = MODNN(k+IPRIM)) {
+      q = 1;
+      for (j = 1; j <= no_eras; j++)
+	if (reg[j] != A0) {
+	  reg[j] = MODNN(reg[j] + j);
+	  q ^= ALPHA_TO[reg[j]];
+	}
+      if (q != 0)
+	continue;
+      /* store root and error location number indices */
+      root[count] = i;
+      loc[count] = k;
+      count++;
+    }
+    if (count != no_eras) {
+      printf("count = %d no_eras = %d\n lambda(x) is WRONG\n",count,no_eras);
+      count = -1;
+      goto finish;
+    }
+#if DEBUG >= 2
+    printf("\n Erasure positions as determined by roots of Eras Loc Poly:\n");
+    for (i = 0; i < count; i++)
+      printf("%d ", loc[i]);
+    printf("\n");
+#endif
+#endif
+  }
+  for(i=0;i<NROOTS+1;i++)
+    //    printf("%d  %d  %d\n",i,lambda[i],INDEX_OF[lambda[i]]);
+    b[i] = INDEX_OF[lambda[i]];
+  
+  /*
+   * Begin Berlekamp-Massey algorithm to determine error+erasure
+   * locator polynomial
+   */
+  r = no_eras;
+  el = no_eras;
+  while (++r <= NROOTS) {	/* r is the step number */
+    /* Compute discrepancy at the r-th step in poly-form */
+    discr_r = 0;
+    for (i = 0; i < r; i++){
+      if ((lambda[i] != 0) && (s[r-i-1] != A0)) {
+	discr_r ^= ALPHA_TO[MODNN(INDEX_OF[lambda[i]] + s[r-i-1])];
+      }
+    }
+    discr_r = INDEX_OF[discr_r];	/* Index form */
+    if (discr_r == A0) {
+      /* 2 lines below: B(x) <-- x*B(x) */
+      memmove(&b[1],b,NROOTS*sizeof(b[0]));
+      b[0] = A0;
+    } else {
+      /* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
+      t[0] = lambda[0];
+      for (i = 0 ; i < NROOTS; i++) {
+	if(b[i] != A0)
+	  t[i+1] = lambda[i+1] ^ ALPHA_TO[MODNN(discr_r + b[i])];
+	else
+	  t[i+1] = lambda[i+1];
+      }
+      if (2 * el <= r + no_eras - 1) {
+	el = r + no_eras - el;
+	/*
+	 * 2 lines below: B(x) <-- inv(discr_r) *
+	 * lambda(x)
+	 */
+	for (i = 0; i <= NROOTS; i++)
+	  b[i] = (lambda[i] == 0) ? A0 : MODNN(INDEX_OF[lambda[i]] - discr_r + NN);
+      } else {
+	/* 2 lines below: B(x) <-- x*B(x) */
+	memmove(&b[1],b,NROOTS*sizeof(b[0]));
+	b[0] = A0;
+      }
+      memcpy(lambda,t,(NROOTS+1)*sizeof(t[0]));
+    }
+  }
+
+  /* Convert lambda to index form and compute deg(lambda(x)) */
+  deg_lambda = 0;
+  for(i=0;i<NROOTS+1;i++){
+    lambda[i] = INDEX_OF[lambda[i]];
+    if(lambda[i] != A0)
+      deg_lambda = i;
+  }
+  /* Find roots of the error+erasure locator polynomial by Chien search */
+  memcpy(&reg[1],&lambda[1],NROOTS*sizeof(reg[0]));
+  count = 0;		/* Number of roots of lambda(x) */
+  for (i = 1,k=IPRIM-1; i <= NN; i++,k = MODNN(k+IPRIM)) {
+    q = 1; /* lambda[0] is always 0 */
+    for (j = deg_lambda; j > 0; j--){
+      if (reg[j] != A0) {
+	reg[j] = MODNN(reg[j] + j);
+	q ^= ALPHA_TO[reg[j]];
+      }
+    }
+    if (q != 0)
+      continue; /* Not a root */
+    /* store root (index-form) and error location number */
+#if DEBUG>=2
+    printf("count %d root %d loc %d\n",count,i,k);
+#endif
+    root[count] = i;
+    loc[count] = k;
+    /* If we've already found max possible roots,
+     * abort the search to save time
+     */
+    if(++count == deg_lambda)
+      break;
+  }
+  if (deg_lambda != count) {
+    /*
+     * deg(lambda) unequal to number of roots => uncorrectable
+     * error detected
+     */
+    count = -1;
+    goto finish;
+  }
+  /*
+   * Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
+   * x**NROOTS). in index form. Also find deg(omega).
+   */
+  deg_omega = deg_lambda-1;
+  for (i = 0; i <= deg_omega;i++){
+    tmp = 0;
+    for(j=i;j >= 0; j--){
+      if ((s[i - j] != A0) && (lambda[j] != A0))
+	tmp ^= ALPHA_TO[MODNN(s[i - j] + lambda[j])];
+    }
+    omega[i] = INDEX_OF[tmp];
+  }
+
+  /*
+   * Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
+   * inv(X(l))**(FCR-1) and den = lambda_pr(inv(X(l))) all in poly-form
+   */
+  for (j = count-1; j >=0; j--) {
+    num1 = 0;
+    for (i = deg_omega; i >= 0; i--) {
+      if (omega[i] != A0)
+	num1  ^= ALPHA_TO[MODNN(omega[i] + i * root[j])];
+    }
+    num2 = ALPHA_TO[MODNN(root[j] * (FCR - 1) + NN)];
+    den = 0;
+    
+    /* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
+    for (i = min(deg_lambda,NROOTS-1) & ~1; i >= 0; i -=2) {
+      if(lambda[i+1] != A0)
+	den ^= ALPHA_TO[MODNN(lambda[i+1] + i * root[j])];
+    }
+#if DEBUG >= 1
+    if (den == 0) {
+      printf("\n ERROR: denominator = 0\n");
+      count = -1;
+      goto finish;
+    }
+#endif
+    /* Apply error to data */
+    if (num1 != 0 && loc[j] >= PAD) {
+      data[loc[j]-PAD] ^= ALPHA_TO[MODNN(INDEX_OF[num1] + INDEX_OF[num2] + NN - INDEX_OF[den])];
+    }
+  }
+ finish:
+  if(eras_pos != NULL){
+    for(i=0;i<count;i++)
+      eras_pos[i] = loc[i];
+  }
+  return count;
+}
diff --git a/lib/superfox/encode_rs.c b/lib/superfox/encode_rs.c
new file mode 100644
index 000000000..11507b373
--- /dev/null
+++ b/lib/superfox/encode_rs.c
@@ -0,0 +1,52 @@
+/* Reed-Solomon encoder
+ * Copyright 2002, Phil Karn, KA9Q
+ * May be used under the terms of the GNU General Public License (GPL)
+ */
+#include <string.h>
+
+#ifdef FIXED
+#include "fixed.h"
+#elif defined(BIGSYM)
+#include "int_sf.h"
+#else
+#include "char.h"
+#endif
+
+void ENCODE_RS(
+#ifdef FIXED
+DTYPE *data, DTYPE *bb,int pad){
+#else
+void *p,DTYPE *data, DTYPE *bb){
+  struct rs *rs = (struct rs *)p;
+#endif
+  int i, j;
+  DTYPE feedback;
+
+#ifdef FIXED
+  /* Check pad parameter for validity */
+  if(pad < 0 || pad >= NN)
+    return;
+#endif
+
+  memset(bb,0,NROOTS*sizeof(DTYPE));
+
+  for(i=0;i<NN-NROOTS-PAD;i++){
+    feedback = INDEX_OF[data[i] ^ bb[0]];
+    if(feedback != A0){      /* feedback term is non-zero */
+#ifdef UNNORMALIZED
+      /* This line is unnecessary when GENPOLY[NROOTS] is unity, as it must
+       * always be for the polynomials constructed by init_rs()
+       */
+      feedback = MODNN(NN - GENPOLY[NROOTS] + feedback);
+#endif
+      for(j=1;j<NROOTS;j++)
+	bb[j] ^= ALPHA_TO[MODNN(feedback + GENPOLY[NROOTS-j])];
+    }
+    /* Shift */
+    memmove(&bb[0],&bb[1],sizeof(DTYPE)*(NROOTS-1));
+    if(feedback != A0)
+      bb[NROOTS-1] = ALPHA_TO[MODNN(feedback + GENPOLY[0])];
+    else
+      bb[NROOTS-1] = 0;
+  }
+}
diff --git a/lib/superfox/init_rs.c b/lib/superfox/init_rs.c
new file mode 100644
index 000000000..a3fa72c30
--- /dev/null
+++ b/lib/superfox/init_rs.c
@@ -0,0 +1,126 @@
+/* Initialize a RS codec
+ *
+ * Copyright 2002 Phil Karn, KA9Q
+ * May be used under the terms of the GNU General Public License (GPL)
+ */
+#include <stdlib.h>
+
+#ifdef CCSDS
+#include "ccsds.h"
+#elif defined(BIGSYM)
+#include "int_sf.h"
+#else
+#include "char.h"
+#endif
+
+#define NULL ((void *)0)
+
+void FREE_RS(void *p){
+  struct rs *rs = (struct rs *)p;
+
+  free(rs->alpha_to);
+  free(rs->index_of);
+  free(rs->genpoly);
+  free(rs);
+}
+
+/* Initialize a Reed-Solomon codec
+ * symsize = symbol size, bits (1-8)
+ * gfpoly = Field generator polynomial coefficients
+ * fcr = first root of RS code generator polynomial, index form
+ * prim = primitive element to generate polynomial roots
+ * nroots = RS code generator polynomial degree (number of roots)
+ * pad = padding bytes at front of shortened block
+ */
+void *INIT_RS(int symsize,int gfpoly,int fcr,int prim,
+	int nroots,int pad){
+  struct rs *rs;
+  int i, j, sr,root,iprim;
+
+  /* Check parameter ranges */
+  if(symsize < 0 || symsize > 8*sizeof(DTYPE))
+    return NULL; /* Need version with ints rather than chars */
+
+  if(fcr < 0 || fcr >= (1<<symsize))
+    return NULL;
+  if(prim <= 0 || prim >= (1<<symsize))
+    return NULL;
+  if(nroots < 0 || nroots >= (1<<symsize))
+    return NULL; /* Can't have more roots than symbol values! */
+  if(pad < 0 || pad >= ((1<<symsize) -1 - nroots))
+    return NULL; /* Too much padding */
+
+  rs = (struct rs *)calloc(1,sizeof(struct rs));
+  rs->mm = symsize;
+  rs->nn = (1<<symsize)-1;
+  rs->pad = pad;
+
+  rs->alpha_to = (DTYPE *)malloc(sizeof(DTYPE)*(rs->nn+1));
+  if(rs->alpha_to == NULL){
+    free(rs);
+    return NULL;
+  }
+  rs->index_of = (DTYPE *)malloc(sizeof(DTYPE)*(rs->nn+1));
+  if(rs->index_of == NULL){
+    free(rs->alpha_to);
+    free(rs);
+    return NULL;
+  }
+
+  /* Generate Galois field lookup tables */
+  rs->index_of[0] = A0; /* log(zero) = -inf */
+  rs->alpha_to[A0] = 0; /* alpha**-inf = 0 */
+  sr = 1;
+  for(i=0;i<rs->nn;i++){
+    rs->index_of[sr] = i;
+    rs->alpha_to[i] = sr;
+    sr <<= 1;
+    if(sr & (1<<symsize))
+      sr ^= gfpoly;
+    sr &= rs->nn;
+  }
+  if(sr != 1){
+    /* field generator polynomial is not primitive! */
+    free(rs->alpha_to);
+    free(rs->index_of);
+    free(rs);
+    return NULL;
+  }
+
+  /* Form RS code generator polynomial from its roots */
+  rs->genpoly = (DTYPE *)malloc(sizeof(DTYPE)*(nroots+1));
+  if(rs->genpoly == NULL){
+    free(rs->alpha_to);
+    free(rs->index_of);
+    free(rs);
+    return NULL;
+  }
+  rs->fcr = fcr;
+  rs->prim = prim;
+  rs->nroots = nroots;
+
+  /* Find prim-th root of 1, used in decoding */
+  for(iprim=1;(iprim % prim) != 0;iprim += rs->nn)
+    ;
+  rs->iprim = iprim / prim;
+
+  rs->genpoly[0] = 1;
+  for (i = 0,root=fcr*prim; i < nroots; i++,root += prim) {
+    rs->genpoly[i+1] = 1;
+
+    /* Multiply rs->genpoly[] by  @**(root + x) */
+    for (j = i; j > 0; j--){
+      if (rs->genpoly[j] != 0)
+	rs->genpoly[j] = rs->genpoly[j-1] ^ rs->alpha_to[modnn(rs,rs->index_of[rs->genpoly[j]] + root)];
+      else
+	rs->genpoly[j] = rs->genpoly[j-1];
+    }
+    /* rs->genpoly[0] can never be zero */
+    rs->genpoly[0] = rs->alpha_to[modnn(rs,rs->index_of[rs->genpoly[0]] + root)];
+  }
+  /* convert rs->genpoly[] to index form for quicker encoding */
+  for (i = 0; i <= nroots; i++)
+    rs->genpoly[i] = rs->index_of[rs->genpoly[i]];
+
+  return rs;
+}
diff --git a/lib/superfox/int_sf.h b/lib/superfox/int_sf.h
new file mode 100644
index 000000000..2b5a805e8
--- /dev/null
+++ b/lib/superfox/int_sf.h
@@ -0,0 +1,56 @@
+/* Include file to configure the RS codec for integer symbols
+ *
+ * Copyright 2002, Phil Karn, KA9Q
+ * May be used under the terms of the GNU General Public License (GPL)
+ */
+#define DTYPE int
+
+/* Reed-Solomon codec control block */
+struct rs {
+  int mm;              /* Bits per symbol */
+  int nn;              /* Symbols per block (= (1<<mm)-1) */
+  DTYPE *alpha_to;     /* log lookup table */
+  DTYPE *index_of;     /* Antilog lookup table */
+  DTYPE *genpoly;      /* Generator polynomial */
+  int nroots;     /* Number of generator roots = number of parity symbols */
+  int fcr;        /* First consecutive root, index form */
+  int prim;       /* Primitive element, index form */
+  int iprim;      /* prim-th root of 1, index form */
+  int pad;        /* Padding bytes in shortened block */
+};
+
+static inline int modnn(struct rs *rs,int x){
+  while (x >= rs->nn) {
+    x -= rs->nn;
+    x = (x >> rs->mm) + (x & rs->nn);
+  }
+  return x;
+}
+#define MODNN(x) modnn(rs,x)
+
+#define MM (rs->mm)
+#define NN (rs->nn)
+#define ALPHA_TO (rs->alpha_to) 
+#define INDEX_OF (rs->index_of)
+#define GENPOLY (rs->genpoly)
+#define NROOTS (rs->nroots)
+#define FCR (rs->fcr)
+#define PRIM (rs->prim)
+#define IPRIM (rs->iprim)
+#define PAD (rs->pad)
+#define A0 (NN)
+
+#define ENCODE_RS encode_rs_sf
+#define DECODE_RS decode_rs_sf
+#define INIT_RS init_rs_sf
+#define FREE_RS free_rs_sf
+
+void ENCODE_RS(void *p,DTYPE *data,DTYPE *parity);
+int DECODE_RS(void *p,DTYPE *data,int *eras_pos,int no_eras);
+void *INIT_RS(int symsize,int gfpoly,int fcr,
+		   int prim,int nroots,int pad);
+void FREE_RS(void *p);
+
+
+
+
diff --git a/lib/superfox/rs_sf.c b/lib/superfox/rs_sf.c
new file mode 100644
index 000000000..972fb2906
--- /dev/null
+++ b/lib/superfox/rs_sf.c
@@ -0,0 +1,57 @@
+#include <stdio.h>
+#include "rs_sf.h"
+
+void *rs;
+static int first=1;
+static int nn,kk,nroots,npad;
+
+void rs_init_(int *mm, int *nq, int *nn0, int *kk0, int *nfz)
+{
+  nn=*nn0;
+  kk=*kk0;
+  nroots=nn-kk;
+  npad=*nq-1-nn;
+  if(*mm==6) rs=init_rs_sf(*mm,0x43,*nfz,1,nroots,npad);   //M=6
+  if(*mm==7) rs=init_rs_sf(*mm,0x89,*nfz,1,nroots,npad);   //M=7
+  if(*mm==8) rs=init_rs_sf(*mm,0x11d,*nfz,1,nroots,npad);  //M=8
+  first=0;
+}
+
+void rs_encode_(int *dgen, int *sent)
+     // Encode JT65 data dgen[...], producing sent[...].
+{
+  int dat1[256];
+  int b[256];
+  int i;
+
+  // Reverse data order for the Karn codec.
+  for(i=0; i<kk; i++) {
+    dat1[i]=dgen[kk-1-i];
+  }
+  // Compute the parity symbols
+  encode_rs_sf(rs,dat1,b);
+
+  // Move parity symbols and data into sent[] array, in reverse order.
+  for (i = 0; i < nroots; i++) sent[nroots-1-i] = b[i];
+  for (i = 0; i < kk; i++) sent[i+nroots] = dat1[kk-1-i];
+}
+
+void rs_decode_(int *recd0, int *era0, int *numera0, int *decoded, int *nerr)
+     // Decode JT65 received data recd0[63], producing decoded[12].
+     // Erasures are indicated in era0[numera].  The number of corrected
+     // errors is *nerr.  If the data are uncorrectable, *nerr=-1 is
+     // returned.
+{
+  int numera;
+  int i;
+  int era_pos[200];
+  int recd[255];
+
+  numera=*numera0;
+  for(i=0; i<kk; i++) recd[i]=recd0[nn-1-i];
+  for(i=0; i<nroots; i++) recd[kk+i]=recd0[nroots-1-i];
+  if(numera) 
+    for(i=0; i<numera; i++) era_pos[i]=era0[i];
+  *nerr=decode_rs_sf(rs,recd,era_pos,numera);
+  for(i=0; i<kk; i++) decoded[i]=recd[kk-1-i];
+}
diff --git a/lib/superfox/rs_sf.h b/lib/superfox/rs_sf.h
new file mode 100644
index 000000000..5b39ae31c
--- /dev/null
+++ b/lib/superfox/rs_sf.h
@@ -0,0 +1,35 @@
+/* User include file for the Reed-Solomon codec
+ * Copyright 2002, Phil Karn KA9Q
+ * May be used under the terms of the GNU General Public License (GPL)
+ */
+
+/* General purpose RS codec, 8-bit symbols */
+void encode_rs_char(void *rs,unsigned char *data,unsigned char *parity);
+int decode_rs_char(void *rs,unsigned char *data,int *eras_pos,
+		   int no_eras);
+void *init_rs_char(int symsize,int gfpoly,
+		   int fcr,int prim,int nroots,
+		   int pad);
+void free_rs_char(void *rs);
+
+/* General purpose RS codec, integer symbols */
+void encode_rs_sf(void *rs,int *data,int *parity);
+int decode_rs_sf(void *rs,int *data,int *eras_pos,int no_eras);
+void *init_rs_sf(int symsize,int gfpoly,int fcr,
+		  int prim,int nroots,int pad);
+void free_rs_sf(void *rs);
+
+/* CCSDS standard (255,223) RS codec with conventional (*not* dual-basis)
+ * symbol representation
+ */
+void encode_rs_8(unsigned char *data,unsigned char *parity,int pad);
+int decode_rs_8(unsigned char *data,int *eras_pos,int no_eras,int pad);
+
+/* CCSDS standard (255,223) RS codec with dual-basis symbol representation */
+void encode_rs_ccsds(unsigned char *data,unsigned char *parity,int pad);
+int decode_rs_ccsds(unsigned char *data,int *eras_pos,int no_eras,int pad);
+
+/* Tables to map from conventional->dual (Taltab) and
+ * dual->conventional (Tal1tab) bases
+ */
+extern unsigned char Taltab[],Tal1tab[];