An asymmetric packing code of radius R is a set of binary n-words so that no binary n-word can be reached from more than

Definitions and mathematical statement

For $x\in\{0,1\}^n$ and an integer $R\ge0$, let \[ B_R^+(x)=\{y\in\{0,1\}^n:x_i\le y_i\text{ for all }i, \ d_H(x,y)\le R\}. \] Thus a channel output is formed by changing at most $R$ zeroes to ones, without any change in the opposite direction. An asymmetric packing code is a set $C\subseteq\{0,1\}^n$ whose upward balls are pairwise disjoint: \[ B_R^+(x)\cap B_R^+(x')=\varnothing \qquad(x,x'\in C,\ x\ne x'). \] Define $A^+(n,R)$ as the largest size of such a code. Determine this maximum, especially its sharp growth for fixed $R$ and $n\to\infty$.

Pairwise disjointness means that every possible received word has at most one codeword that could have produced it with the allowed errors; it does not require every word to be reachable. Equivalently, two distinct codewords $x,x'$ cannot both have at most $R$ ones outside the other's support. Indeed, their coordinatewise union is the smallest possible common output. This gives a concrete pairwise feasibility criterion for codes.

The objective is maximization under disjointness. The word "covering" in the last sentence of the collected source is a slip: the preceding definition and this entry concern packing.

Short English statement

Determine the largest binary code whose upward asymmetric radius-R balls are pairwise disjoint.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100075 (AMR-030-0075), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Words and Codes, source-order bullet 75. Original source indexed by AMR-030-0075. https://people.math.sc.edu/cooper/combprob.html.