{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R658",
  "content_hash": null,
  "slug": "rm28-artifact-full-quadratic-c-crosscheck",
  "type": "artifact",
  "title": "Full 2^21-quadratic C cross-check of the cubic distance",
  "summary": "A third exact implementation enumerates every homogeneous quadratic directly and decodes both slices against all 256 affine functions.",
  "relevance": "For Covering radius of the second-order Reed-Muller code RM(2,8), record rm28-artifact-full-quadratic-c-crosscheck (“Full 2^21-quadratic C cross-check of the cubic distance”) supplies evidence or a replay used to check the packet. The record states: A third exact implementation enumerates every homogeneous quadratic directly and decodes both slices against all 256 affine functions.",
  "relevance_source": "recorded",
  "body": "Join source_lines with LF, append a terminal LF, and save the result as `rm28_full.c` in a disposable directory. The recorded command compiles and runs a clean C11 implementation. It visits all \\(2^{21}\\) homogeneous quadratics by Gray code and compares each of the two 7-variable slices with every affine function. This route uses no quotient reduction, row reduction, Walsh transform, or Python code. Its full-space histogram is exactly 256 times the quotient histogram in rm28-artifact-exact-cubic-distance, which independently checks the eight-dimensional invariance multiplicity as well as the minimum 88.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the displayed cubic across every homogeneous quadratic on each 7-variable slice and every affine slice correction",
    "bounds": {
      "slice_variables": {
        "min": 7,
        "max": 7
      },
      "slices": {
        "min": 2,
        "max": 2
      },
      "homogeneous_quadratics": {
        "min": 2097152,
        "max": 2097152
      },
      "affine_masks_per_slice": {
        "min": 256,
        "max": 256
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "complete",
    "kind": "inline_c11_full_quadratic_crosscheck",
    "command": "cc -O3 -std=c11 -Wall -Wextra rm28_full.c -o rm28_full && ./rm28_full",
    "entrypoint": "Join source_lines with LF, append one terminal LF, and save as rm28_full.c",
    "runtime": "Apple clang 21.0.0 (clang-2100.0.123.102), arm64-apple-darwin25.2.0",
    "citation": {
      "locator": "Self-contained C11 program authored and executed on 2026-07-28"
    },
    "dependencies": [
      {
        "name": "Apple clang",
        "version": "21.0.0 (clang-2100.0.123.102)",
        "license": "Apache-2.0 WITH LLVM-exception"
      }
    ],
    "outputs": {
      "source_sha256": "effa78eee5e183a168ea3eaefa1d8b625004b06351af81afae0a4d6ec8d54433",
      "stdout_sha256": "f27c922f54801ad02ebe9fc9326d7b54010d0ec93c655b5dde4a26fab056c9bc",
      "expected_stdout": "algorithm=full-quadratic-gray-code-and-exhaustive-affine-masks\nhomogeneous_quadratics=2097152\naffine_masks_per_slice=256\nhistogram=88:7168,92:260096,96:759808,100:774144,104:279552,108:14336,112:2048\nquotient_histogram_multiplicity=256\nminimum=88\n",
      "homogeneous_quadratics": 2097152,
      "affine_masks_per_slice": 256,
      "minimum": 88,
      "quotient_histogram_multiplicity": 256,
      "histogram": {
        "88": 7168,
        "92": 260096,
        "96": 759808,
        "100": 774144,
        "104": 279552,
        "108": 14336,
        "112": 2048
      }
    },
    "runtime_seconds": 0.784239,
    "inline_source": [
      "#include <inttypes.h>",
      "#include <stdint.h>",
      "#include <stdio.h>",
      "",
      "typedef struct {",
      "    uint64_t low;",
      "    uint64_t high;",
      "} Mask;",
      "",
      "static const int PAIRS[21][2] = {",
      "    {0, 1}, {0, 2}, {0, 3}, {0, 4}, {0, 5}, {0, 6}, {1, 2},",
      "    {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5},",
      "    {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6},",
      "};",
      "static const int G_TERMS[4][3] = {",
      "    {0, 2, 6}, {0, 3, 5}, {1, 3, 6}, {2, 3, 4},",
      "};",
      "static const int P_TERMS[4][2] = {",
      "    {0, 1}, {2, 5}, {4, 6}, {5, 6},",
      "};",
      "static const uint64_t EXPECTED[7] = {",
      "    7168, 260096, 759808, 774144, 279552, 14336, 2048,",
      "};",
      "",
      "static Mask pair_truth[21];",
      "static Mask affine_truth[256];",
      "",
      "static void set_bit(Mask *mask, int position) {",
      "    if (position < 64) {",
      "        mask->low |= UINT64_C(1) << position;",
      "    } else {",
      "        mask->high |= UINT64_C(1) << (position - 64);",
      "    }",
      "}",
      "",
      "static Mask xor_mask(Mask left, Mask right) {",
      "    Mask result = {left.low ^ right.low, left.high ^ right.high};",
      "    return result;",
      "}",
      "",
      "static int weight(Mask mask) {",
      "    return __builtin_popcountll(mask.low) + __builtin_popcountll(mask.high);",
      "}",
      "",
      "static int affine_distance(Mask mask) {",
      "    int best = 128;",
      "    for (int i = 0; i < 256; ++i) {",
      "        int candidate = weight(xor_mask(mask, affine_truth[i]));",
      "        if (candidate < best) {",
      "            best = candidate;",
      "        }",
      "    }",
      "    return best;",
      "}",
      "",
      "int main(void) {",
      "    Mask g = {0, 0};",
      "    Mask p = {0, 0};",
      "    for (int x = 0; x < 128; ++x) {",
      "        for (int k = 0; k < 21; ++k) {",
      "            if (((x >> PAIRS[k][0]) & 1) && ((x >> PAIRS[k][1]) & 1)) {",
      "                set_bit(&pair_truth[k], x);",
      "            }",
      "        }",
      "        int g_value = 0;",
      "        int p_value = 0;",
      "        for (int k = 0; k < 4; ++k) {",
      "            g_value ^= ((x >> G_TERMS[k][0]) & 1)",
      "                & ((x >> G_TERMS[k][1]) & 1)",
      "                & ((x >> G_TERMS[k][2]) & 1);",
      "            p_value ^= ((x >> P_TERMS[k][0]) & 1)",
      "                & ((x >> P_TERMS[k][1]) & 1);",
      "        }",
      "        if (g_value) {",
      "            set_bit(&g, x);",
      "        }",
      "        if (p_value) {",
      "            set_bit(&p, x);",
      "        }",
      "        for (int coefficients = 0; coefficients < 256; ++coefficients) {",
      "            int value = (coefficients >> 7) & 1;",
      "            value ^= __builtin_popcount((unsigned)(coefficients & 127 & x)) & 1;",
      "            if (value) {",
      "                set_bit(&affine_truth[coefficients], x);",
      "            }",
      "        }",
      "    }",
      "",
      "    uint64_t histogram[129] = {0};",
      "    Mask h = {0, 0};",
      "    unsigned previous_gray = 0;",
      "    for (unsigned index = 0; index < (1u << 21); ++index) {",
      "        unsigned gray = index ^ (index >> 1);",
      "        if (index) {",
      "            int changed = __builtin_ctz(gray ^ previous_gray);",
      "            h = xor_mask(h, pair_truth[changed]);",
      "        }",
      "        previous_gray = gray;",
      "        int score = affine_distance(xor_mask(g, h));",
      "        score += affine_distance(xor_mask(xor_mask(g, p), h));",
      "        ++histogram[score];",
      "    }",
      "",
      "    uint64_t total = 0;",
      "    for (int i = 0; i < 7; ++i) {",
      "        int score = 88 + 4 * i;",
      "        if (histogram[score] != EXPECTED[i]) {",
      "            return 2;",
      "        }",
      "        total += histogram[score];",
      "    }",
      "    if (total != (1u << 21)) {",
      "        return 2;",
      "    }",
      "",
      "    puts(\"algorithm=full-quadratic-gray-code-and-exhaustive-affine-masks\");",
      "    puts(\"homogeneous_quadratics=2097152\");",
      "    puts(\"affine_masks_per_slice=256\");",
      "    puts(\"histogram=88:7168,92:260096,96:759808,100:774144,104:279552,108:14336,112:2048\");",
      "    puts(\"quotient_histogram_multiplicity=256\");",
      "    puts(\"minimum=88\");",
      "    return 0;",
      "}"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Self-contained C11 program authored and executed on 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R663",
      "title": "An eight-term cubic has exact second-order nonlinearity 88",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "reed-muller-rm2-8-covering-radius",
      "title": "reed muller rm2 8 covering radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}