The code you describe is constructed by encoding each element of a -element set (with even) as a binary vector, where each coordinate corresponds to an unordered partition of the set into two equal-sized subsets. The vector's bit at a given partition is 1 if the element belongs to one subset of the partition and 0 otherwise (with an arbitrary but consistent choice between the subset and its...

What you are describing is the incidence code coming from the complete 2-(k, k/2, λ) design (the “complete design” on a k‑point set with all k/2‑subsets as blocks), with complements identified. In the language of association schemes this sits in the Johnson scheme J(k, k/2): take one column for each unordered complementary pair {S, S^c} with |S| = k/2, and for message i ∈ [k] put a 1 in that colum...

Yes, the code is already well–known. In the language of design and coding theory it is simply the binary incidence code of the Johnson design

  J(k, k/2) (k even).

Construction (design point of view)
• Take the point set P = {1,…,k}.
• As “blocks’’ take all k/2–subsets of P, but regard a subset and its complement as the same b...

What you have written down is nothing mysterious – it is exactly the “point‐versus‐block’’ incidence code of the 2‐design whose blocks are the fifty‐fifty splits of a k‐element set. In the language of design theory it is the point‐code of the 2–(v,k/2,λ) design with
 v = k,
 block‐size = k/2,
 number of blocks b = ( k choose k/2 ),
 replication number r = ( k−1 choose k/2−1 ),
 and λ = ...

What you obtain is nothing new – it is the binary code that is obtained from the classical Johnson scheme J(k,k/2). In the design / coding–theory literature it is usually called

 • the point– (or incidence-) code of the Johnson design,
 • the Johnson code J(k,1), or, in Delsarte’s terminology, C(J(k,k/2)).

Construction in design language
Let V={1,…,k} with k even and let 𝔅 be the fam...

Excellent question! You've described a very natural and important combinatorial object that appears in several fields. Your intuition that it has been studied before is correct.

Here's a breakdown of what this code is called, its properties, and where to find references.

Names and Contexts

The most appropriate name depends on the context (PCP theory, coding theory, or combinatorics).

1. *...

Identifying the Balanced Variant of the Long Code

Based on the description, you're looking for a coding scheme that has the following properties:

1. It encodes a set of k values using binary strings with one coordinate for each partition of the k values into two equal-sized subsets
2. For each partition, you arbitrarily choose one of the two possible binary functions
3. This results in a code ...

This code you're describing is known as the First-Order Reed-Muller code punctured at a specific set of positions, or more specifically, it's closely related to what's sometimes called the Hadamard code restricted to balanced functions.

Let me explain the connection:

1. The standard long code encodes a value by the truth table of all Boolean functions $f: {0,1}^k \to {...