Best Known (11, 17, s)-Nets in Base 81
(11, 17, 177149)-Net over F81 — Constructive and digital
Digital (11, 17, 177149)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 177149, F81, 6, 6) (dual of [(177149, 6), 1062877, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8117, 531447, F81, 6) (dual of [531447, 531430, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 531448, F81, 6) (dual of [531448, 531431, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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 Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 531448, F81, 6) (dual of [531448, 531431, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(8117, 531447, F81, 6) (dual of [531447, 531430, 7]-code), using
(11, 17, 531449)-Net over F81 — Digital
Digital (11, 17, 531449)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 531449, F81, 6) (dual of [531449, 531432, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(817, 8, F81, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,81)), using
- dual of repetition code with length 8 [i]
- linear OA(811, 8, F81, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, 81, F81, 1) (dual of [81, 80, 2]-code), using
- Reed–Solomon code RS(80,81) [i]
- discarding factors / shortening the dual code based on linear OA(811, 81, F81, 1) (dual of [81, 80, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
(11, 17, large)-Net in Base 81 — Upper bound on s
There is no (11, 17, large)-net in base 81, because
- 4 times m-reduction [i] would yield (11, 13, large)-net in base 81, but