Best Known (84, 84+24, s)-Nets in Base 9
(84, 84+24, 4921)-Net over F9 — Constructive and digital
Digital (84, 108, 4921)-net over F9, using
- 92 times duplication [i] based on digital (82, 106, 4921)-net over F9, using
- net defined by OOA [i] based on linear OOA(9106, 4921, F9, 24, 24) (dual of [(4921, 24), 117998, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(9106, 59052, F9, 24) (dual of [59052, 58946, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(9106, 59054, F9, 24) (dual of [59054, 58948, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(9101, 59049, F9, 23) (dual of [59049, 58948, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(9106, 59054, F9, 24) (dual of [59054, 58948, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(9106, 59052, F9, 24) (dual of [59052, 58946, 25]-code), using
- net defined by OOA [i] based on linear OOA(9106, 4921, F9, 24, 24) (dual of [(4921, 24), 117998, 25]-NRT-code), using
(84, 84+24, 49515)-Net over F9 — Digital
Digital (84, 108, 49515)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9108, 49515, F9, 24) (dual of [49515, 49407, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(9108, 59062, F9, 24) (dual of [59062, 58954, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(996, 59049, F9, 22) (dual of [59049, 58953, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(991, 59049, F9, 21) (dual of [59049, 58958, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(9108, 59062, F9, 24) (dual of [59062, 58954, 25]-code), using
(84, 84+24, large)-Net in Base 9 — Upper bound on s
There is no (84, 108, large)-net in base 9, because
- 22 times m-reduction [i] would yield (84, 86, large)-net in base 9, but