RAID-Like Solution with Custom Parity Levels

Formulas for Parity Calculation

The formulas for calculating parity levels p₁, p₂, and p₃ are defined as follows:

Data Values

Each device holds one byte of data:

D₁ = 0x01, D₂ = 0x02, D₃ = 0x03, D₄ = 0x04, D₅ = 0x05, D₆ = 0x06

Parity Calculations

p₁ Calculation

Step-by-step XOR computation:

0x01 ⊕ 0x02 = 0x03
0x03 ⊕ 0x03 = 0x00
0x00 ⊕ 0x04 = 0x04
0x04 ⊕ 0x05 = 0x01
0x01 ⊕ 0x06 = 0x07

Result: p₁ = 0x07

p₂ Calculation

Using GF(2⁸) multiplication:

2 ⋅ D₁ = 0x02, 2 ⋅ D₂ = 0x04, 2 ⋅ D₃ = 0x06,
2 ⋅ D₄ = 0x08, 2 ⋅ D₅ = 0x0A, 2 ⋅ D₆ = 0x0C

Step-by-step XOR computation:
0x02 ⊕ 0x04 = 0x06
0x06 ⊕ 0x06 = 0x00
0x00 ⊕ 0x08 = 0x08
0x08 ⊕ 0x0A = 0x02
0x02 ⊕ 0x0C = 0x0E

Result: p₂ = 0x0E

p₃ Calculation

Using GF(2⁸) multiplication:

4 ⋅ D₁ = 0x04, 4 ⋅ D₂ = 0x08, 4 ⋅ D₃ = 0x0C,
4 ⋅ D₄ = 0x10, 4 ⋅ D₅ = 0x14, 4 ⋅ D₆ = 0x18

Step-by-step XOR computation:
0x04 ⊕ 0x08 = 0x0C
0x0C ⊕ 0x0C = 0x00
0x00 ⊕ 0x10 = 0x10
0x10 ⊕ 0x14 = 0x04
0x04 ⊕ 0x18 = 0x1C

Result: p₃ = 0x1C

Simulating Data Recovery

Assume device D₄ is missing. Using p₁, p₂, and p₃, we recover D₄:

Recovery Using p₁

Formula:
D₄ = p₁ ⊕ D₁ ⊕ D₂ ⊕ D₃ ⊕ D₅ ⊕ D₆

Step-by-step:
0x01 ⊕ 0x02 = 0x03
0x03 ⊕ 0x03 = 0x00
0x00 ⊕ 0x05 = 0x05
0x05 ⊕ 0x06 = 0x03
0x03 ⊕ 0x07 = 0x04

Recovered D₄: 0x04

Final Results

Original Data: D₁ = 0x01, D₂ = 0x02, D₃ = 0x03, D₄ = 0x04, D₅ = 0x05, D₆ = 0x06

Parity Blocks: p₁ = 0x07, p₂ = 0x0E, p₃ = 0x1C

Recovered Device: D₄ = 0x04