**Introduction**

Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the weight distribution of $C$.
Often, the weight distribution is represented in the form of a polynomial, the *weight enumerator*. The well-known *MacWilliams identities* relate the weight enumerator of a code $C$ to the weight enumerator of the *dual code* $C^\perp$.

Suppose that we do not know $A_i^C$.
Instead, let $C$ be an element from a set of length-$n$ codes $\mathcal{C}_n$, for which we only know the *averaged asymptotic weight spectrum*, (the average is over the set $\mathcal{C}_n$)

$$S^{\mathcal{C}_\infty}_\delta := \lim_{n \rightarrow \infty} \frac{1}{n} \log \frac{1}{|\mathcal{C_n}|} \sum_{C \in \mathcal{C}_n} A^C_{ \lfloor \delta n \rfloor }$$

where $ 0 \leq \delta \leq 1$ represents the *relative weight*.

Now, the question is to find a MacWilliams-type of identity that relates the *dual (averaged asymptotic) spectrum* $S^{\mathcal{C}^\perp_\infty}_\delta$, where $\mathcal{C}_n^\perp$ is the set of codes dual to codes in $\mathcal{C}_n$, to the (known) weight spectrum $S_\delta^{\mathcal{C}_\infty}$.

**Motivation**

For large-length codes with many codewords, it is often infeasible to determine the weight distribution $A_i$. On the other hand, the averaged weight distribution is known for some families of codes.
As an example, take Low-Density Parity-Check (LDPC) codes. These are linear codes that are randomly constructed by sampling the parity check matrix according to some particular strategy. While it is often hard to make non-trivial statements about individual members of this set (in random-code terminology, the set is usually called the *ensemble*), various results are known for the ensemble, including weight spectras [1].

From a practical point of view, LDPC codes have several desirable properties that make them well-suited for use in real-life applications, such as telecommunication and storage products. To determine whether these codes are also suited for certain cryptological applications, more insight is required into the weight spectrum of the dual ensemble.

An alternative approach is to compute the dual spectrum numerically, by first evaluating the asymptotic weight distribution on a grid of equispaced points, and then using the original discrete MacWilliams identities. This approach, however, is numerically unstable because these computations involve enormous values due to the binomial coefficients.

**More about the MacWilliams Identities**

We define the enumerator polynomial of the code $C$ as

$$ W(C; x, y) := \sum_i A^C_i x^i y ^{n-i} $$

Then, the MacWilliams identity is given by

$$ W(C^\perp; x,y) = \frac{1}{|C|} W(C ; y-x ,y+x).$$

Another form of the MacWilliams identity is the following,

$$ \hspace{10em} A_d^{C^\perp} = \frac{1}{|C|} \sum_{i=0}^n A_i^C P_d(i;n), \hspace{10em} (1)$$

where $P_d(i;n)$ is a *Krawtchouk polynomial*,

$$ P_d(i;n):=\sum_{j=0}^d \binom{i}{j}\binom{n-i}{d-j} (-1)^j $$

**Informal Conjecture**

Informally speaking, the latter form of the MacWilliams identity, (1), looks like a transform, e.g. as a Fourier series. So it seems natural to conjecture that it should be possible to take the limit for $n\rightarrow \infty$ in eq. (1) yielding some integral, similar to how Fourier extended the Fourier series to the continuous Fourier transform. Unfortunately, the author from this MathOverflow question got stuck here. Other references that might help in attacking this problem are [2] (partly about asymptotic behaviour of Krawtchouck polynomials) and work of Krasikov [3] on Krawtchouk polynomials.

**References**

[1] S. Litsyn and V. Shevelev, "On Ensembles of Low-Density Parity Check Codes: Asymptotic Distance Distributions," *IEEE Trans. Inf. Theory*, April 2002.

[2] G. Kalai and N. Linial, "On the Distance Distribution of Codes," *IEEE Trans. Inf. Theory*, September 1995.

[3] http://www.brunel.ac.uk/about/acad/siscm/maths/people/acad/IliaKrasikov