How does random noise in the digital world typically look?

Suppose you have a memory of n bits, and suppose that a "random noise" hits the memory in such a way that the probability of each bit being affected is at most t.

What will be the typical behavior of such a random digital noise? Part of the question is ** to define "random noise" in the best way possible**, and then answer it for this definition.

The same question can be asked for a memory based on a larger r-letters alphabet.

In particular (as Greg mentioned) I am interested to know: ** Will the "random noise" behave typically in a way which is close to be independent on the different bits? or will it be highly correlated? ** or pehaps the answer depends on the size of the alphabet and the value of t?

The source of this question is an easy fact about quantum memory which asserts that if you consider a random noise operation acting on a quantum memory with n qubits, and if the probability that every qubit is damaged is a tiny real number t, then typically the noise has the following form: with large probability nothing happens and with tiny probability a large fraction of qubits are harmed. (So in this case, random noise is highly correlated.)

I made one try for the digital (binary) question but I am not sure at all that it is the "correct" definition for what "random noise" is. I considered a random permutation of all the 2^n binary strings of length n subject to the condition that at most t bits are flipped. I will be happy if somebody can think about the case of larger alphabet and also offers the conceptually "correct" framework for this question.

Perhaps the best way to ask this question (following Jason's and Greg's remarks) is this: What is the appropriate way to define a "random" probability distribution on {0,1}^n and on {0,1,...,r}^n.

Given such a definition we the would like to understand the typical behavior of such a random probability distribution with the additional property that the probability for the ith coordinate to be non zero is at most t for every i.

**Updated formulation:** One way to solve the question was proposed by Greg: The space of stochastic maps on a finite probability space is a convex polytope. So you can choose one using standard Euclidean measure. (In our case, the finite probability space is the uniform probability space on {0,1}^n, and more generally, the uniform probability space on {0,1,...,r}^n.)

The question is (stated for the binary case) this: Let t>0 be a small real number. How does a random stochastic map, conditioned on the property that a proportion of t bits are corrupted looks like. Is it also the case, as Greg expects, that (like in the quantum case,) with large probability nothing happens and with small probability many bits are corrupted.

I will be happy to see how to do the calculation precisely or heuristically, or perhaps to find it in the literature.

**Update:** The notion of Arikan's **polar coding** (See e.g. this paper) is somewhat related to this question.

How Quantum Computers Can Failis broken, here is a replacement: arxiv.org/abs/quant-ph/0607021 $\endgroup$