+==========================
+NAND Error-correction Code
+==========================
+
Introduction
============
Back to ecc.
Let's give a small figure:
+========= ==== ==== ==== ==== ==== ==== ==== ==== === === === === ====
byte 0: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp2 rp4 ... rp14
byte 1: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp2 rp4 ... rp14
byte 2: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp3 rp4 ... rp14
byte 3: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp3 rp4 ... rp14
byte 4: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp2 rp5 ... rp14
-....
+...
byte 254: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp3 rp5 ... rp15
byte 255: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp3 rp5 ... rp15
cp1 cp0 cp1 cp0 cp1 cp0 cp1 cp0
cp3 cp3 cp2 cp2 cp3 cp3 cp2 cp2
cp5 cp5 cp5 cp5 cp4 cp4 cp4 cp4
+========= ==== ==== ==== ==== ==== ==== ==== ==== === === === === ====
This figure represents a sector of 256 bytes.
cp is my abbreviation for column parity, rp for row parity.
Let's start to explain column parity.
-cp0 is the parity that belongs to all bit0, bit2, bit4, bit6.
-so the sum of all bit0, bit2, bit4 and bit6 values + cp0 itself is even.
+
+- cp0 is the parity that belongs to all bit0, bit2, bit4, bit6.
+
+ so the sum of all bit0, bit2, bit4 and bit6 values + cp0 itself is even.
+
Similarly cp1 is the sum of all bit1, bit3, bit5 and bit7.
-cp2 is the parity over bit0, bit1, bit4 and bit5
-cp3 is the parity over bit2, bit3, bit6 and bit7.
-cp4 is the parity over bit0, bit1, bit2 and bit3.
-cp5 is the parity over bit4, bit5, bit6 and bit7.
+
+- cp2 is the parity over bit0, bit1, bit4 and bit5
+- cp3 is the parity over bit2, bit3, bit6 and bit7.
+- cp4 is the parity over bit0, bit1, bit2 and bit3.
+- cp5 is the parity over bit4, bit5, bit6 and bit7.
+
Note that each of cp0 .. cp5 is exactly one bit.
Row parity actually works almost the same.
-rp0 is the parity of all even bytes (0, 2, 4, 6, ... 252, 254)
-rp1 is the parity of all odd bytes (1, 3, 5, 7, ..., 253, 255)
-rp2 is the parity of all bytes 0, 1, 4, 5, 8, 9, ...
-(so handle two bytes, then skip 2 bytes).
-rp3 is covers the half rp2 does not cover (bytes 2, 3, 6, 7, 10, 11, ...)
-for rp4 the rule is cover 4 bytes, skip 4 bytes, cover 4 bytes, skip 4 etc.
-so rp4 calculates parity over bytes 0, 1, 2, 3, 8, 9, 10, 11, 16, ...)
-and rp5 covers the other half, so bytes 4, 5, 6, 7, 12, 13, 14, 15, 20, ..
+
+- rp0 is the parity of all even bytes (0, 2, 4, 6, ... 252, 254)
+- rp1 is the parity of all odd bytes (1, 3, 5, 7, ..., 253, 255)
+- rp2 is the parity of all bytes 0, 1, 4, 5, 8, 9, ...
+ (so handle two bytes, then skip 2 bytes).
+- rp3 is covers the half rp2 does not cover (bytes 2, 3, 6, 7, 10, 11, ...)
+- for rp4 the rule is cover 4 bytes, skip 4 bytes, cover 4 bytes, skip 4 etc.
+
+ so rp4 calculates parity over bytes 0, 1, 2, 3, 8, 9, 10, 11, 16, ...)
+- and rp5 covers the other half, so bytes 4, 5, 6, 7, 12, 13, 14, 15, 20, ..
+
The story now becomes quite boring. I guess you get the idea.
-rp6 covers 8 bytes then skips 8 etc
-rp7 skips 8 bytes then covers 8 etc
-rp8 covers 16 bytes then skips 16 etc
-rp9 skips 16 bytes then covers 16 etc
-rp10 covers 32 bytes then skips 32 etc
-rp11 skips 32 bytes then covers 32 etc
-rp12 covers 64 bytes then skips 64 etc
-rp13 skips 64 bytes then covers 64 etc
-rp14 covers 128 bytes then skips 128
-rp15 skips 128 bytes then covers 128
+
+- rp6 covers 8 bytes then skips 8 etc
+- rp7 skips 8 bytes then covers 8 etc
+- rp8 covers 16 bytes then skips 16 etc
+- rp9 skips 16 bytes then covers 16 etc
+- rp10 covers 32 bytes then skips 32 etc
+- rp11 skips 32 bytes then covers 32 etc
+- rp12 covers 64 bytes then skips 64 etc
+- rp13 skips 64 bytes then covers 64 etc
+- rp14 covers 128 bytes then skips 128
+- rp15 skips 128 bytes then covers 128
In the end the parity bits are grouped together in three bytes as
follows:
+
+===== ===== ===== ===== ===== ===== ===== ===== =====
ECC Bit 7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0
+===== ===== ===== ===== ===== ===== ===== ===== =====
ECC 0 rp07 rp06 rp05 rp04 rp03 rp02 rp01 rp00
ECC 1 rp15 rp14 rp13 rp12 rp11 rp10 rp09 rp08
ECC 2 cp5 cp4 cp3 cp2 cp1 cp0 1 1
+===== ===== ===== ===== ===== ===== ===== ===== =====
I detected after writing this that ST application note AN1823
(http://www.st.com/stonline/) gives a much
nicer picture.(but they use line parity as term where I use row parity)
Oh well, I'm graphically challenged, so suffer with me for a moment :-)
+
And I could not reuse the ST picture anyway for copyright reasons.
=========
Implementing the parity calculation is pretty simple.
-In C pseudocode:
-for (i = 0; i < 256; i++)
-{
+In C pseudocode::
+
+ for (i = 0; i < 256; i++)
+ {
if (i & 0x01)
rp1 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp1;
else
cp3 = bit7 ^ bit6 ^ bit3 ^ bit2 ^ cp3
cp4 = bit3 ^ bit2 ^ bit1 ^ bit0 ^ cp4
cp5 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ cp5
-}
+ }
Analysis 0
Attempt 1
=========
-const char parity[256] = {
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0
-};
-
-void ecc1(const unsigned char *buf, unsigned char *code)
-{
- int i;
- const unsigned char *bp = buf;
- unsigned char cur;
- unsigned char rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7;
- unsigned char rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15;
- unsigned char par;
-
- par = 0;
- rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0;
- rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0;
- rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0;
- rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0;
-
- for (i = 0; i < 256; i++)
- {
- cur = *bp++;
- par ^= cur;
- if (i & 0x01) rp1 ^= cur; else rp0 ^= cur;
- if (i & 0x02) rp3 ^= cur; else rp2 ^= cur;
- if (i & 0x04) rp5 ^= cur; else rp4 ^= cur;
- if (i & 0x08) rp7 ^= cur; else rp6 ^= cur;
- if (i & 0x10) rp9 ^= cur; else rp8 ^= cur;
- if (i & 0x20) rp11 ^= cur; else rp10 ^= cur;
- if (i & 0x40) rp13 ^= cur; else rp12 ^= cur;
- if (i & 0x80) rp15 ^= cur; else rp14 ^= cur;
- }
- code[0] =
- (parity[rp7] << 7) |
- (parity[rp6] << 6) |
- (parity[rp5] << 5) |
- (parity[rp4] << 4) |
- (parity[rp3] << 3) |
- (parity[rp2] << 2) |
- (parity[rp1] << 1) |
- (parity[rp0]);
- code[1] =
- (parity[rp15] << 7) |
- (parity[rp14] << 6) |
- (parity[rp13] << 5) |
- (parity[rp12] << 4) |
- (parity[rp11] << 3) |
- (parity[rp10] << 2) |
- (parity[rp9] << 1) |
- (parity[rp8]);
- code[2] =
- (parity[par & 0xf0] << 7) |
- (parity[par & 0x0f] << 6) |
- (parity[par & 0xcc] << 5) |
- (parity[par & 0x33] << 4) |
- (parity[par & 0xaa] << 3) |
- (parity[par & 0x55] << 2);
- code[0] = ~code[0];
- code[1] = ~code[1];
- code[2] = ~code[2];
-}
+::
+
+ const char parity[256] = {
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1,
+ 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0
+ };
+
+ void ecc1(const unsigned char *buf, unsigned char *code)
+ {
+ int i;
+ const unsigned char *bp = buf;
+ unsigned char cur;
+ unsigned char rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7;
+ unsigned char rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15;
+ unsigned char par;
+
+ par = 0;
+ rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0;
+ rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0;
+ rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0;
+ rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0;
+
+ for (i = 0; i < 256; i++)
+ {
+ cur = *bp++;
+ par ^= cur;
+ if (i & 0x01) rp1 ^= cur; else rp0 ^= cur;
+ if (i & 0x02) rp3 ^= cur; else rp2 ^= cur;
+ if (i & 0x04) rp5 ^= cur; else rp4 ^= cur;
+ if (i & 0x08) rp7 ^= cur; else rp6 ^= cur;
+ if (i & 0x10) rp9 ^= cur; else rp8 ^= cur;
+ if (i & 0x20) rp11 ^= cur; else rp10 ^= cur;
+ if (i & 0x40) rp13 ^= cur; else rp12 ^= cur;
+ if (i & 0x80) rp15 ^= cur; else rp14 ^= cur;
+ }
+ code[0] =
+ (parity[rp7] << 7) |
+ (parity[rp6] << 6) |
+ (parity[rp5] << 5) |
+ (parity[rp4] << 4) |
+ (parity[rp3] << 3) |
+ (parity[rp2] << 2) |
+ (parity[rp1] << 1) |
+ (parity[rp0]);
+ code[1] =
+ (parity[rp15] << 7) |
+ (parity[rp14] << 6) |
+ (parity[rp13] << 5) |
+ (parity[rp12] << 4) |
+ (parity[rp11] << 3) |
+ (parity[rp10] << 2) |
+ (parity[rp9] << 1) |
+ (parity[rp8]);
+ code[2] =
+ (parity[par & 0xf0] << 7) |
+ (parity[par & 0x0f] << 6) |
+ (parity[par & 0xcc] << 5) |
+ (parity[par & 0x33] << 4) |
+ (parity[par & 0xaa] << 3) |
+ (parity[par & 0x55] << 2);
+ code[0] = ~code[0];
+ code[1] = ~code[1];
+ code[2] = ~code[2];
+ }
Still pretty straightforward. The last three invert statements are there to
give a checksum of 0xff 0xff 0xff for an empty flash. In an empty flash
Attempt 2
=========
-extern const char parity[256];
-
-void ecc2(const unsigned char *buf, unsigned char *code)
-{
- int i;
- const unsigned long *bp = (unsigned long *)buf;
- unsigned long cur;
- unsigned long rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7;
- unsigned long rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15;
- unsigned long par;
-
- par = 0;
- rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0;
- rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0;
- rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0;
- rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0;
-
- for (i = 0; i < 64; i++)
- {
- cur = *bp++;
- par ^= cur;
- if (i & 0x01) rp5 ^= cur; else rp4 ^= cur;
- if (i & 0x02) rp7 ^= cur; else rp6 ^= cur;
- if (i & 0x04) rp9 ^= cur; else rp8 ^= cur;
- if (i & 0x08) rp11 ^= cur; else rp10 ^= cur;
- if (i & 0x10) rp13 ^= cur; else rp12 ^= cur;
- if (i & 0x20) rp15 ^= cur; else rp14 ^= cur;
- }
- /*
- we need to adapt the code generation for the fact that rp vars are now
- long; also the column parity calculation needs to be changed.
- we'll bring rp4 to 15 back to single byte entities by shifting and
- xoring
- */
- rp4 ^= (rp4 >> 16); rp4 ^= (rp4 >> 8); rp4 &= 0xff;
- rp5 ^= (rp5 >> 16); rp5 ^= (rp5 >> 8); rp5 &= 0xff;
- rp6 ^= (rp6 >> 16); rp6 ^= (rp6 >> 8); rp6 &= 0xff;
- rp7 ^= (rp7 >> 16); rp7 ^= (rp7 >> 8); rp7 &= 0xff;
- rp8 ^= (rp8 >> 16); rp8 ^= (rp8 >> 8); rp8 &= 0xff;
- rp9 ^= (rp9 >> 16); rp9 ^= (rp9 >> 8); rp9 &= 0xff;
- rp10 ^= (rp10 >> 16); rp10 ^= (rp10 >> 8); rp10 &= 0xff;
- rp11 ^= (rp11 >> 16); rp11 ^= (rp11 >> 8); rp11 &= 0xff;
- rp12 ^= (rp12 >> 16); rp12 ^= (rp12 >> 8); rp12 &= 0xff;
- rp13 ^= (rp13 >> 16); rp13 ^= (rp13 >> 8); rp13 &= 0xff;
- rp14 ^= (rp14 >> 16); rp14 ^= (rp14 >> 8); rp14 &= 0xff;
- rp15 ^= (rp15 >> 16); rp15 ^= (rp15 >> 8); rp15 &= 0xff;
- rp3 = (par >> 16); rp3 ^= (rp3 >> 8); rp3 &= 0xff;
- rp2 = par & 0xffff; rp2 ^= (rp2 >> 8); rp2 &= 0xff;
- par ^= (par >> 16);
- rp1 = (par >> 8); rp1 &= 0xff;
- rp0 = (par & 0xff);
- par ^= (par >> 8); par &= 0xff;
-
- code[0] =
- (parity[rp7] << 7) |
- (parity[rp6] << 6) |
- (parity[rp5] << 5) |
- (parity[rp4] << 4) |
- (parity[rp3] << 3) |
- (parity[rp2] << 2) |
- (parity[rp1] << 1) |
- (parity[rp0]);
- code[1] =
- (parity[rp15] << 7) |
- (parity[rp14] << 6) |
- (parity[rp13] << 5) |
- (parity[rp12] << 4) |
- (parity[rp11] << 3) |
- (parity[rp10] << 2) |
- (parity[rp9] << 1) |
- (parity[rp8]);
- code[2] =
- (parity[par & 0xf0] << 7) |
- (parity[par & 0x0f] << 6) |
- (parity[par & 0xcc] << 5) |
- (parity[par & 0x33] << 4) |
- (parity[par & 0xaa] << 3) |
- (parity[par & 0x55] << 2);
- code[0] = ~code[0];
- code[1] = ~code[1];
- code[2] = ~code[2];
-}
+::
+
+ extern const char parity[256];
+
+ void ecc2(const unsigned char *buf, unsigned char *code)
+ {
+ int i;
+ const unsigned long *bp = (unsigned long *)buf;
+ unsigned long cur;
+ unsigned long rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7;
+ unsigned long rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15;
+ unsigned long par;
+
+ par = 0;
+ rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0;
+ rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0;
+ rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0;
+ rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0;
+
+ for (i = 0; i < 64; i++)
+ {
+ cur = *bp++;
+ par ^= cur;
+ if (i & 0x01) rp5 ^= cur; else rp4 ^= cur;
+ if (i & 0x02) rp7 ^= cur; else rp6 ^= cur;
+ if (i & 0x04) rp9 ^= cur; else rp8 ^= cur;
+ if (i & 0x08) rp11 ^= cur; else rp10 ^= cur;
+ if (i & 0x10) rp13 ^= cur; else rp12 ^= cur;
+ if (i & 0x20) rp15 ^= cur; else rp14 ^= cur;
+ }
+ /*
+ we need to adapt the code generation for the fact that rp vars are now
+ long; also the column parity calculation needs to be changed.
+ we'll bring rp4 to 15 back to single byte entities by shifting and
+ xoring
+ */
+ rp4 ^= (rp4 >> 16); rp4 ^= (rp4 >> 8); rp4 &= 0xff;
+ rp5 ^= (rp5 >> 16); rp5 ^= (rp5 >> 8); rp5 &= 0xff;
+ rp6 ^= (rp6 >> 16); rp6 ^= (rp6 >> 8); rp6 &= 0xff;
+ rp7 ^= (rp7 >> 16); rp7 ^= (rp7 >> 8); rp7 &= 0xff;
+ rp8 ^= (rp8 >> 16); rp8 ^= (rp8 >> 8); rp8 &= 0xff;
+ rp9 ^= (rp9 >> 16); rp9 ^= (rp9 >> 8); rp9 &= 0xff;
+ rp10 ^= (rp10 >> 16); rp10 ^= (rp10 >> 8); rp10 &= 0xff;
+ rp11 ^= (rp11 >> 16); rp11 ^= (rp11 >> 8); rp11 &= 0xff;
+ rp12 ^= (rp12 >> 16); rp12 ^= (rp12 >> 8); rp12 &= 0xff;
+ rp13 ^= (rp13 >> 16); rp13 ^= (rp13 >> 8); rp13 &= 0xff;
+ rp14 ^= (rp14 >> 16); rp14 ^= (rp14 >> 8); rp14 &= 0xff;
+ rp15 ^= (rp15 >> 16); rp15 ^= (rp15 >> 8); rp15 &= 0xff;
+ rp3 = (par >> 16); rp3 ^= (rp3 >> 8); rp3 &= 0xff;
+ rp2 = par & 0xffff; rp2 ^= (rp2 >> 8); rp2 &= 0xff;
+ par ^= (par >> 16);
+ rp1 = (par >> 8); rp1 &= 0xff;
+ rp0 = (par & 0xff);
+ par ^= (par >> 8); par &= 0xff;
+
+ code[0] =
+ (parity[rp7] << 7) |
+ (parity[rp6] << 6) |
+ (parity[rp5] << 5) |
+ (parity[rp4] << 4) |
+ (parity[rp3] << 3) |
+ (parity[rp2] << 2) |
+ (parity[rp1] << 1) |
+ (parity[rp0]);
+ code[1] =
+ (parity[rp15] << 7) |
+ (parity[rp14] << 6) |
+ (parity[rp13] << 5) |
+ (parity[rp12] << 4) |
+ (parity[rp11] << 3) |
+ (parity[rp10] << 2) |
+ (parity[rp9] << 1) |
+ (parity[rp8]);
+ code[2] =
+ (parity[par & 0xf0] << 7) |
+ (parity[par & 0x0f] << 6) |
+ (parity[par & 0xcc] << 5) |
+ (parity[par & 0x33] << 4) |
+ (parity[par & 0xaa] << 3) |
+ (parity[par & 0x55] << 2);
+ code[0] = ~code[0];
+ code[1] = ~code[1];
+ code[2] = ~code[2];
+ }
The parity array is not shown any more. Note also that for these
examples I kinda deviated from my regular programming style by allowing
Attempt 3
=========
-Odd replaced:
- if (i & 0x01) rp5 ^= cur; else rp4 ^= cur;
- if (i & 0x02) rp7 ^= cur; else rp6 ^= cur;
- if (i & 0x04) rp9 ^= cur; else rp8 ^= cur;
- if (i & 0x08) rp11 ^= cur; else rp10 ^= cur;
- if (i & 0x10) rp13 ^= cur; else rp12 ^= cur;
- if (i & 0x20) rp15 ^= cur; else rp14 ^= cur;
-with
- if (i & 0x01) rp5 ^= cur;
- if (i & 0x02) rp7 ^= cur;
- if (i & 0x04) rp9 ^= cur;
- if (i & 0x08) rp11 ^= cur;
- if (i & 0x10) rp13 ^= cur;
- if (i & 0x20) rp15 ^= cur;
-
- and outside the loop added:
- rp4 = par ^ rp5;
- rp6 = par ^ rp7;
- rp8 = par ^ rp9;
- rp10 = par ^ rp11;
- rp12 = par ^ rp13;
- rp14 = par ^ rp15;
+Odd replaced::
+
+ if (i & 0x01) rp5 ^= cur; else rp4 ^= cur;
+ if (i & 0x02) rp7 ^= cur; else rp6 ^= cur;
+ if (i & 0x04) rp9 ^= cur; else rp8 ^= cur;
+ if (i & 0x08) rp11 ^= cur; else rp10 ^= cur;
+ if (i & 0x10) rp13 ^= cur; else rp12 ^= cur;
+ if (i & 0x20) rp15 ^= cur; else rp14 ^= cur;
+
+with::
+
+ if (i & 0x01) rp5 ^= cur;
+ if (i & 0x02) rp7 ^= cur;
+ if (i & 0x04) rp9 ^= cur;
+ if (i & 0x08) rp11 ^= cur;
+ if (i & 0x10) rp13 ^= cur;
+ if (i & 0x20) rp15 ^= cur;
+
+and outside the loop added::
+
+ rp4 = par ^ rp5;
+ rp6 = par ^ rp7;
+ rp8 = par ^ rp9;
+ rp10 = par ^ rp11;
+ rp12 = par ^ rp13;
+ rp14 = par ^ rp15;
And after that the code takes about 30% more time, although the number of
statements is reduced. This is also reflected in the assembly code.
=========
Unrolled the loop 1, 2, 3 and 4 times.
-For 4 the code starts with:
+For 4 the code starts with::
for (i = 0; i < 4; i++)
{
==========
Unrolling once gains about 15%
+
Unrolling twice keeps the gain at about 15%
+
Unrolling three times gives a gain of 30% compared to attempt 2.
+
Unrolling four times gives a marginal improvement compared to unrolling
three times.
Effectively so all odd digit rp assignments in the loop were removed.
This included the else clause of the if statements.
-Of course after the loop we need to correct things by adding code like:
+Of course after the loop we need to correct things by adding code like::
+
rp5 = par ^ rp4;
+
Also the initial assignments (rp5 = 0; etc) could be removed.
Along the line I also removed the initialisation of rp0/1/2/3.
Attempt 6
=========
-THe code within the for loop was changed to:
+THe code within the for loop was changed to::
for (i = 0; i < 4; i++)
{
definitely seemed to be the jackpot!
There is a little bit more room for improvement though. There are three
-places with statements:
-rp4 ^= cur; rp6 ^= cur;
+places with statements::
+
+ rp4 ^= cur; rp6 ^= cur;
+
It seems more efficient to also maintain a variable rp4_6 in the while
loop; This eliminates 3 statements per loop. Of course after the loop we
-need to correct by adding:
- rp4 ^= rp4_6;
- rp6 ^= rp4_6
+need to correct by adding::
+
+ rp4 ^= rp4_6;
+ rp6 ^= rp4_6
+
Furthermore there are 4 sequential assignments to rp8. This can be
encoded slightly more efficiently by saving tmppar before those 4 lines
and later do rp8 = rp8 ^ tmppar ^ notrp8;
Attempt 7
=========
-The new code now looks like:
+The new code now looks like::
for (i = 0; i < 4; i++)
{
further there is still room to optimize the generation of the ecc codes.
We can simply calculate the total parity. If this is 0 then rp4 = rp5
etc. If the parity is 1, then rp4 = !rp5;
+
But if rp4 = rp5 we do not need rp5 etc. We can just write the even bits
-in the result byte and then do something like
+in the result byte and then do something like::
+
code[0] |= (code[0] << 1);
+
Lets test this.
kind of other things, like having dedicated parity arrays to avoid the
shift after parity[rp7] << 7; No gain.
Change the lookup using the parity array by using shift operators (e.g.
-replace parity[rp7] << 7 with:
-rp7 ^= (rp7 << 4);
-rp7 ^= (rp7 << 2);
-rp7 ^= (rp7 << 1);
-rp7 &= 0x80;
+replace parity[rp7] << 7 with::
+
+ rp7 ^= (rp7 << 4);
+ rp7 ^= (rp7 << 2);
+ rp7 ^= (rp7 << 1);
+ rp7 &= 0x80;
+
No gain.
The only marginal change was inverting the parity bits, so we can remove
For correcting errors I again used the ST application note as a starter,
but I also peeked at the existing code.
+
The algorithm itself is pretty straightforward. Just xor the given and
the calculated ecc. If all bytes are 0 there is no problem. If 11 bits
are 1 we have one correctable bit error. If there is 1 bit 1, we have an
error in the given ecc code.
+
It proved to be fastest to do some table lookups. Performance gain
introduced by this is about a factor 2 on my system when a repair had to
be done, and 1% or so if no repair had to be done.
+
Code size increased from 330 bytes to 686 bytes for this function.
(gcc 4.2, -O3)
The gain when calculating the ecc is tremendous. Om my development hardware
a speedup of a factor of 18 for ecc calculation was achieved. On a test on an
embedded system with a MIPS core a factor 7 was obtained.
+
On a test with a Linksys NSLU2 (ARMv5TE processor) the speedup was a factor
5 (big endian mode, gcc 4.1.2, -O3)
+
For correction not much gain could be obtained (as bitflips are rare). Then
again there are also much less cycles spent there.
this is very tricky (at least for intel hw).
Author: Frans Meulenbroeks
+
Copyright (C) 2008 Koninklijke Philips Electronics NV.