70046: Added Error correction

Author: OliverKillane

70046 - Advanced Digital Systems Design/Error Correction.md

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+## Fault Sources
+- Radiation (alpha particles hitting silicon and inducing enough charge to cause a bit flip)
+- Hardware breakage (by manufacturing or over lifetime due to aging)
+## Error Detection
+### Parity Bits
+Adding a single bit to a set of bits that represents the number of `1`s.
+- Can detect an odd number of fault events (as this changes the parity bit)
+
+We can include multiple check parity bits to allow error correction (to deduce which bit was flipped, given a single fault).
+```rust
+P0 = parity(D0, D1, D3)
+P1 = parity(D0, D2, D3)
+P2 = parity(D1, D2, D3)
+
+// If only one of P0, P1, P2 is unchanged, then we can deduce which data bit was changed
+```
+Each data bit needs to affect a different combination of check bits to allow us to infer the data bit changed, from the check bit change.
+$$2^n -n - 1 \ \text{data bits are covered by} \ n \ \text{ parity bits}$$
+- $n$ bits means $2^n$ combinations
+- of the $2^n$ combinations, only $n$ affect a single bit (`...010...`)
+- one combination should change no bits (no parity change)
+```python
+def parity(bits: list[bool]) -> bool:
+    """ Parity as the number of 1s is odd=True, even=False """
+    return bool(sum(bits) % 2)
+
+def gen_indexes(size: int) -> list[set[int]]:
+    """ Generate the parity indexes for `size` data bits
+        [index{parity indices... }, ]
+        - If all parities chan ge except index, then index was flipped.
+        - The set contains the bit indicies to calculate parity
+    """
+    return [{j for j in range(size) if j != i} for i in range(size)]
+
+def gen_parity_bits(data_bits: list[bool]) -> list[bool]:
+    """ Get the error correction bits """
+    return [parity([data_bits[i] for i in s]) for s in gen_indexes(len(data_bits))]
+  
+def check(data_bits: list[bool], parity_bits: list[bool]) -> int | None | str:
+    """ Compares the parity
+    single fault => (all except one parity bit are flipped) then 
+                    return the data bit index flipped
+    string       => double fault
+    none         => no change
+    """
+    same_index: int | None = None
+    differing: bool = False
+    for i, (o, n) in enumerate(zip(parity_bits, gen_parity_bits(data_bits))):
+        if o == n:
+            if same_index is not None and differing:
+                return "Double fault"
+            same_index = i
+        else:
+            differing = True
+    return same_index
+
+original_bits = [True, True, False, False, True]
+original_parity = gen_parity_bits(original_bits)
+new_bits = [True, False, False, False, True]
+new_parity = gen_parity_bits(new_bits)
+assert check(new_bits, original_parity) == 1
+```
+
+We can also add an extra bit fir the parity of the parity bits (to detect double faults)
+### Basic Checksum
+Add all data words together (ignoring carry) and store result.
+- Easy to compute
+- Errors can cancel out (no error detected)
+### Modified Checksum
+On every carry, add a 1 instead, this way carry information is not entirely lost.
+- Errors can still cancel out
+### Cyclic Redundancy Check
+Use polynomial division
+[[TODO]]