Recovery balance is preserved by duality

More strongly, for every encoded coordinate $i$ of a linear code $C$ of length $n\geq1$, $$a_i(C)+a_i(C^\perp)=n,$$ where $a_i$ is the expected number of independent uniform reads with replacement needed to recover that coordinate. Consequently, the recovery profile is constant for $C$ exactly when it is constant for $C^\perp$, proving the conjecture. The argument pairs complementary observation sets, writes expectations as weighted sums over sets, and uses equal complementary weights. A coordinate known to be zero has recovery time zero. The written note also handles the convention requiring one read for such a coordinate: balance equivalence remains valid, but the coordinatewise sum requires a correction. Nonuniform sampling is outside this identity.

More strongly, for every encoded coordinate $i$ of a linear code $C$ of length $n\geq1$, $$a_i(C)+a_i(C^\perp)=n,$$ where $a_i$ is the expected number of independent uniform reads with replacement needed to recover that coordinate. Consequently, the recovery profile is constant for $C$ exactly when it is constant for $C^\perp$, proving the conjecture. The argument pairs complementary observation sets, writes expectations as weighted sums over sets, and uses equal complementary weights. A coordinate known to be zero has recovery time zero. The written note also handles the convention requiring one read for such a coordinate: balance equivalence remains valid, but the coordinatewise sum requires a correction. Nonuniform sampling is outside this identity.