{
  "schema": 1,
  "checks": 478565,
  "field_label_convention": "base-p coefficient vectors; q-polynomial index i means X^(q^i)",
  "field_cases": [
    {
      "p": 2,
      "modulus": [
        1,
        1,
        0,
        1
      ],
      "q": 2,
      "extension_degree": 3,
      "n": 3,
      "k": 1,
      "g": [
        1,
        2,
        4
      ],
      "nonzero_codeword_rank_counts": {
        "3": 7
      },
      "rank_one_ball_size": 50,
      "covered_received_words": 400,
      "ambient_words": 512,
      "received_scope": "all ambient words",
      "tested_received_words": 512,
      "erasure_interpolations": 24,
      "statuses": {
        "DECODED": 400,
        "NO_CODEWORD_WITHIN_RADIUS": 112
      },
      "failure_reasons": {
        "nonzero_formal_remainder": 56,
        "inconsistent_system": 56
      }
    },
    {
      "p": 2,
      "modulus": [
        1,
        1,
        0,
        0,
        1
      ],
      "q": 2,
      "extension_degree": 4,
      "n": 4,
      "k": 2,
      "g": [
        1,
        2,
        4,
        8
      ],
      "nonzero_codeword_rank_counts": {
        "3": 225,
        "4": 30
      },
      "rank_one_ball_size": 226,
      "covered_received_words": 57856,
      "ambient_words": 65536,
      "received_scope": "all ambient words",
      "tested_received_words": 65536,
      "erasure_interpolations": 1536,
      "statuses": {
        "DECODED": 57856,
        "NO_CODEWORD_WITHIN_RADIUS": 7680
      },
      "failure_reasons": {
        "nonzero_formal_remainder": 3840,
        "inconsistent_system": 3840
      }
    },
    {
      "p": 2,
      "modulus": [
        1,
        1,
        0,
        0,
        1
      ],
      "q": 2,
      "extension_degree": 4,
      "n": 3,
      "k": 1,
      "g": [
        1,
        2,
        4
      ],
      "nonzero_codeword_rank_counts": {
        "3": 15
      },
      "rank_one_ball_size": 106,
      "covered_received_words": 1696,
      "ambient_words": 4096,
      "received_scope": "all ambient words",
      "tested_received_words": 4096,
      "erasure_interpolations": 48,
      "statuses": {
        "DECODED": 1696,
        "NO_CODEWORD_WITHIN_RADIUS": 2400
      },
      "failure_reasons": {
        "nonzero_formal_remainder": 2160,
        "inconsistent_system": 240
      }
    },
    {
      "p": 3,
      "modulus": [
        1,
        2,
        0,
        1
      ],
      "q": 3,
      "extension_degree": 3,
      "n": 3,
      "k": 1,
      "g": [
        1,
        3,
        9
      ],
      "nonzero_codeword_rank_counts": {
        "3": 26
      },
      "rank_one_ball_size": 339,
      "covered_received_words": 9153,
      "ambient_words": 19683,
      "received_scope": "all ambient words",
      "tested_received_words": 19683,
      "erasure_interpolations": 81,
      "statuses": {
        "DECODED": 9153,
        "NO_CODEWORD_WITHIN_RADIUS": 10530
      },
      "failure_reasons": {
        "nonzero_formal_remainder": 9828,
        "inconsistent_system": 702
      }
    },
    {
      "p": 2,
      "modulus": [
        1,
        1,
        0,
        0,
        0,
        0,
        1
      ],
      "q": 4,
      "extension_degree": 3,
      "n": 3,
      "k": 1,
      "g": [
        1,
        2,
        4
      ],
      "nonzero_codeword_rank_counts": {
        "3": 63
      },
      "rank_one_ball_size": 1324,
      "covered_received_words": 84736,
      "ambient_words": 262144,
      "received_scope": "deterministic sample: all rank-one errors of zero, every 47th covered word, 64 ambient probes",
      "tested_received_words": 3160,
      "erasure_interpolations": 192,
      "statuses": {
        "DECODED": 3114,
        "NO_CODEWORD_WITHIN_RADIUS": 46
      },
      "failure_reasons": {
        "nonzero_formal_remainder": 45,
        "inconsistent_system": 1
      }
    }
  ],
  "radius_two_promised_cases": 4096,
  "nonprime_base_terminal": {
    "status": "DECODED",
    "message": [
      2
    ],
    "codeword": [
      2,
      4,
      8
    ],
    "error": [
      8,
      8,
      8
    ],
    "error_rank": 1,
    "N": [
      48,
      16
    ],
    "lambda": [
      24,
      1
    ],
    "remainder": [
      0
    ],
    "auxiliary_free_dimension": 0
  },
  "main": {
    "field": "F2[a]/(a^4+a+1)",
    "g": [
      1,
      2,
      4,
      8
    ],
    "message": [
      2,
      1
    ],
    "codeword": [
      3,
      0,
      11,
      15
    ],
    "received": [
      11,
      8,
      3,
      7
    ],
    "interpolation_matrix": [
      [
        1,
        1,
        1,
        11
      ],
      [
        2,
        4,
        3,
        8
      ],
      [
        4,
        3,
        5,
        3
      ],
      [
        8,
        12,
        15,
        7
      ]
    ],
    "right_hand_side": [
      9,
      12,
      5,
      6
    ],
    "rref": [
      [
        1,
        0,
        0,
        0,
        3
      ],
      [
        0,
        1,
        0,
        0,
        12
      ],
      [
        0,
        0,
        1,
        0,
        1
      ],
      [
        0,
        0,
        0,
        1,
        8
      ]
    ],
    "pivots": [
      0,
      1,
      2,
      3
    ],
    "decoded": {
      "status": "DECODED",
      "message": [
        2,
        1
      ],
      "codeword": [
        3,
        0,
        11,
        15
      ],
      "error": [
        8,
        8,
        8,
        8
      ],
      "error_rank": 1,
      "N": [
        3,
        12,
        1
      ],
      "lambda": [
        8,
        1
      ],
      "remainder": [
        0
      ],
      "auxiliary_free_dimension": 0
    },
    "zero_error_auxiliary_choices": 16,
    "zero_error_particular": [
      0,
      4,
      1,
      0
    ],
    "zero_error_null_basis": [
      [
        2,
        1,
        0,
        1
      ]
    ],
    "failure_received": [
      0,
      0,
      1,
      3
    ],
    "failure": {
      "status": "NO_CODEWORD_WITHIN_RADIUS",
      "reason": "nonzero_formal_remainder",
      "N": [
        7,
        1,
        6
      ],
      "lambda": [
        0,
        1
      ],
      "remainder": [
        7
      ],
      "auxiliary_free_dimension": 0
    },
    "outside_promise_received_codeword": [
      3,
      0,
      11,
      15
    ],
    "outside_promise_actual_error_rank": 3
  }
}
