Best Known (83, 111, s)-Nets in Base 9
(83, 111, 937)-Net over F9 — Constructive and digital
Digital (83, 111, 937)-net over F9, using
- 91 times duplication [i] based on digital (82, 110, 937)-net over F9, using
- net defined by OOA [i] based on linear OOA(9110, 937, F9, 28, 28) (dual of [(937, 28), 26126, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(9110, 13118, F9, 28) (dual of [13118, 13008, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 13122, F9, 28) (dual of [13122, 13012, 29]-code), using
- trace code [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- trace code [i] based on linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 13122, F9, 28) (dual of [13122, 13012, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(9110, 13118, F9, 28) (dual of [13118, 13008, 29]-code), using
- net defined by OOA [i] based on linear OOA(9110, 937, F9, 28, 28) (dual of [(937, 28), 26126, 29]-NRT-code), using
(83, 111, 13128)-Net over F9 — Digital
Digital (83, 111, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9111, 13128, F9, 28) (dual of [13128, 13017, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9110, 13126, F9, 28) (dual of [13126, 13016, 29]-code), using
- trace code [i] based on linear OA(8155, 6563, F81, 28) (dual of [6563, 6508, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- trace code [i] based on linear OA(8155, 6563, F81, 28) (dual of [6563, 6508, 29]-code), using
- linear OA(9110, 13127, F9, 27) (dual of [13127, 13017, 28]-code), using Gilbert–Varšamov bound and bm = 9110 > Vbs−1(k−1) = 86 200676 142235 012215 619549 199577 703251 856613 280998 578887 011642 182925 923461 766876 807866 643992 501811 981809 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9110, 13126, F9, 28) (dual of [13126, 13016, 29]-code), using
- construction X with Varšamov bound [i] based on
(83, 111, large)-Net in Base 9 — Upper bound on s
There is no (83, 111, large)-net in base 9, because
- 26 times m-reduction [i] would yield (83, 85, large)-net in base 9, but