Best Known (38−13, 38, s)-Nets in Base 81
(38−13, 38, 88574)-Net over F81 — Constructive and digital
Digital (25, 38, 88574)-net over F81, using
- net defined by OOA [i] based on linear OOA(8138, 88574, F81, 13, 13) (dual of [(88574, 13), 1151424, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8138, 531445, F81, 13) (dual of [531445, 531407, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 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
- discarding factors / shortening the dual code based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8138, 531445, F81, 13) (dual of [531445, 531407, 14]-code), using
(38−13, 38, 265724)-Net over F81 — Digital
Digital (25, 38, 265724)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8138, 265724, F81, 2, 13) (dual of [(265724, 2), 531410, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8138, 531448, F81, 13) (dual of [531448, 531410, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 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
- discarding factors / shortening the dual code based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- OOA 2-folding [i] based on linear OA(8138, 531448, F81, 13) (dual of [531448, 531410, 14]-code), using
(38−13, 38, large)-Net in Base 81 — Upper bound on s
There is no (25, 38, large)-net in base 81, because
- 11 times m-reduction [i] would yield (25, 27, large)-net in base 81, but