One bit extra per cell, in flash memories.

This method reduces the noise by one bit, in flash memories and similar.

Flash memory works fine with 4 levels per cell, but gets troublesome with 16 levels per cell due to noise.

This methods 16 levels will be as solid as 8 levels, since it can stand twice the noise.

Is it precisely 1 bit extra per cell?

Mathematically: Yes.
The value is precisely one bit extra, as a limit when the levels approach infinity.

Practically: Yes.
One extra real bit is stored in each cell.

Noise, or information theoretically: More or less 1 bit.
It depends on the shape of the noise, the errors. There is also a small loss due to edge effects of the stored analog values.

So, what is the noise tolerance?

In these simulations, 8 and 16 level cells are used, as they are representative of todays technology. (2009) The method works better the more levels there are.

Spike noise

The analog range for a cell is set to [0,1]

Size of minimum spike that gives error:
8 Level cell: 0.071455
16 Levels, this system: 0.0644530925
16 Level cell: 0.0333333

These were the results of simulating flash memory with spike errors.

As can be seen, this system has twice the noise tolerance that a 16 level cell have, while keeping 16 levels. That is, 1 bit extra.

The small difference from the 8 level spike is due to edge effects on the analog range.

Gaussian noise

This figure shows the noise tolerance as a function of gaussian noise, in standard deviations.

As one can see, this method approaches the reliability of 8 level cells, because the red line approaches the blue at the bottom.


If there is noise, decoding is estimated to take about 10 operations per bit.
If there is little noise, decoding could be about 100 times faster.
Some operations can be analog.

Encoding should be on the order of 100 times faster than noisy decoding.

Error correction

Since this method give burst errors when the noise is too large, it can favourably be combined with error correction codes built for that, like Reed-Solomon.


If interested, contact me:

Snailmail address:

Kim Øyhus
Odvar Solbergs vei 90, L.8013
N-0970 Oslo


Figure finished. Proof read.

First draft