A \((16,8,5)\) covering is a family of 8-subsets of a 16-point set such that every 5-subset lies in at least one block. The covering number \(C(16,8,5)\) is the smallest possible family size.

The La Jolla Covering Repository page accessed on 2026-07-24 reports \[ 104\le C(16,8,5)\le115. \] It credits the displayed 115-block cover to Alessandro Jurcovich and labels the lower-bound method `Schonheim`. The companion replay verifies every block and all \(\binom{16}{5}=4368\) coverage requirements. The lower-bound record spells out the recursive calculation. Together they certify the interval. They leave an eleven-block gap, so the candidate's request for a single exact value is still open.

The repository supplies 115 distinct, increasing 8-subsets of \(\{1,\ldots,16\}\). Exhaustive enumeration gives coverage multiplicities \[ 1:3124,\quad2:768,\quad3:224,\quad4:192,\quad5:32,\quad6:16,\quad7:12. \] Their sum is 6440, agreeing with the independent incidence check \(115\binom85=6440\). Every requirement has positive multiplicity, which proves \(C(16,8,5)\le115\).

Each block contains between 12 and 32 private 5-subsets, meaning requirements covered by that block alone within this family. All 115 blocks are therefore essential to this particular cover. This local irredundancy makes no minimum-size claim. The canonical newline-delimited block family has SHA-256 digest `52698bf4869691120b55da7419b340005acc8b8a807aafbdeabc3377f4b73664`.

For every covering number, \[ C(v,k,t)\ge \left\lceil\frac vk C(v-1,k-1,t-1)\right\rceil. \] To see this, fix any point. The blocks containing it, with that point deleted, cover all \((t-1)\)-subsets of the other \(v-1\) points. Every point therefore occurs in at least \(C(v-1,k-1,t-1)\) blocks. Summing point-block incidences gives the displayed inequality.

The LJCR base record gives the exact value \(C(13,5,2)=10\), with its lower-bound method listed as `Bounds for fixed number of blocks`. Three recursive applications now give \[ \begin{aligned} C(14,6,3)&\ge\left\lceil\frac{14}{6}\,10\right\rceil=24,\\ C(15,7,4)&\ge\left\lceil\frac{15}{7}\,24\right\rceil=52,\\ C(16,8,5)&\ge\left\lceil\frac{16}{8}\,52\right\rceil=104. \end{aligned} \] The executable record replays the integer arithmetic. The mathematical input at the base of the chain is the sourced exact value 10.

Save the 115-block family displayed by the cited La Jolla Covering Repository page as a local plain-text file, one block per line, and pass its path to the checker. The program canonicalizes whitespace and checks the block-list SHA-256 digest before computing. The published block list is not stored in this packet.

The program checks 115 distinct 8-blocks, enumerates all 4368 five-subsets in lexicographic order, and counts containments. It verifies the full multiplicity distribution and confirms that every displayed block has a private requirement. It also replays the three ceiling calculations leading to 104. The canonical requirement list and `requirement:count` vector receive separate digests. The eight-line standard output has SHA-256 digest `d9e00a088652bec19ab4de0ab0b8ab91e4892d7eea6cb9170d9e86fa55809e79`.