Best Known (63, 76, s)-Nets in Base 7
(63, 76, 39218)-Net over F7 — Constructive and digital
Digital (63, 76, 39218)-net over F7, using
- net defined by OOA [i] based on linear OOA(776, 39218, F7, 13, 13) (dual of [(39218, 13), 509758, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(776, 235309, F7, 13) (dual of [235309, 235233, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(776, 235309, F7, 13) (dual of [235309, 235233, 14]-code), using
(63, 76, 235314)-Net over F7 — Digital
Digital (63, 76, 235314)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(776, 235314, F7, 13) (dual of [235314, 235238, 14]-code), using
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
(63, 76, large)-Net in Base 7 — Upper bound on s
There is no (63, 76, large)-net in base 7, because
- 11 times m-reduction [i] would yield (63, 65, large)-net in base 7, but