Data Encoding Schemes

Data Encoding Schemes

Direct Binary Mapping

Simple 13-bit Encoding

The most straightforward encoding splits the 13 bits between θ and φ parameters:

• Bits 0-6: θ parameter (7 bits = 128 levels)
• Bits 7-12: φ parameter (6 bits = 64 levels)

Gray Code Mapping

Improved Encoding for Error Resilience

Gray code minimizes bit errors during readback by ensuring adjacent values differ by only one bit.

Binary → Gray Code → Physical State

Example:
Binary 1010 → Gray 1111
When reading back 1110 (one bit error in Gray):
Gray 1110 → Binary 1001 (adjacent value)

Benefit: Single-bit read errors only cause ±1 LSB value error

Implementation

Error Mitigation:

• Reduces impact of threshold variations
• Minimizes multi-bit errors
• Better suited for analog storage

Error Correction Coding (ECC)

Reed-Solomon (13,9) Code

Provides robust error correction for multi-level cells:

• Data: 9 bits
• Parity: 4 bits
• Correction capability: 2 symbols (errors)
• Storage efficiency: 69%

Implementation

Reed-Solomon (13,9) Code:

Raw: 13 bits/cell
With ECC: 9 data bits/cell
Still 3× better than 3-bit MLC NAND!

Two-Tier ECC Architecture

Tier 1: Per-Cell Parity (Detect)
- Store checksum of 13 bits in adjacent reference cell
- Detects single-cell failures

Tier 2: Block-Level Reed-Solomon (Correct)
- 64 cells = 832 data bits + 128 parity bits
- Corrects up to 4 cell errors per block

Storage Efficiency:

• Raw: 832 bits / 64 cells = 13 bits/cell
• With ECC: 832 bits data / 72 cells = 11.6 bits/cell useful
• Still excellent density!

Probabilistic Encoding

For AI/ML Applications

Store probability distributions directly in analog values:

Neural Network Weight Storage

Advantages:
- Native analog storage matches analog computation
- No quantization loss
- Perfect for in-memory computing

Gradient Storage

Benefits:
- Smooth gradient representation
- No saturation issues
- Efficient backpropagation

Comparison Table

Encoding Scheme	Efficiency	Error Resilience	Complexity	Best For
Direct Binary	100%	Low	Very Low	Simple applications
Gray Code	100%	Medium	Low	Analog storage
Reed-Solomon ECC	69%	High	Medium	Reliable storage
Probabilistic	Variable	Medium	High	AI/ML workloads

Error Sources and Mitigation

Write Errors

• Source: Incorrect voltage/pulse width
• Mitigation: Verify-write operation
• Success rate: 99.9% first-pass, 99.999% with retry

Read Errors

• Source: Sense amplifier noise, reference drift
• Mitigation: Gray coding + ECC
• BER: < 10⁻¹⁵ with ECC

Retention Errors

• Source: Slow drift over time (years)
• Mitigation: Periodic scrubbing + ECC
• Refresh rate: Once per month typical

Endurance Degradation

• Source: Material fatigue after 10⁶ cycles
• Mitigation: Wear leveling algorithms
• Lifetime: Extended to decades

Recommended Encoding Strategy

General Purpose Storage:

• Use Gray Code mapping for raw data
• Apply Reed-Solomon (13,9) ECC
• Result: 11.6 bits/cell with high reliability

AI/ML Applications:

• Use Probabilistic encoding for weights
• Optional light ECC (parity only)
• Result: Maximum analog fidelity

Ultra-Reliable Storage:

• Use Direct Binary with heavy ECC
• Apply BCH or LDPC codes
• Result: 8-10 bits/cell, enterprise-grade reliability