Best Known (53, 68, s)-Nets in Base 9
(53, 68, 8437)-Net over F9 — Constructive and digital
Digital (53, 68, 8437)-net over F9, using
- 91 times duplication [i] based on digital (52, 67, 8437)-net over F9, using
- net defined by OOA [i] based on linear OOA(967, 8437, F9, 15, 15) (dual of [(8437, 15), 126488, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(956, 59049, F9, 13) (dual of [59049, 58993, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(967, 59060, F9, 15) (dual of [59060, 58993, 16]-code), using
- net defined by OOA [i] based on linear OOA(967, 8437, F9, 15, 15) (dual of [(8437, 15), 126488, 16]-NRT-code), using
(53, 68, 58652)-Net over F9 — Digital
Digital (53, 68, 58652)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(968, 58652, F9, 15) (dual of [58652, 58584, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(968, 59062, F9, 15) (dual of [59062, 58994, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(956, 59049, F9, 13) (dual of [59049, 58993, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(968, 59062, F9, 15) (dual of [59062, 58994, 16]-code), using
(53, 68, large)-Net in Base 9 — Upper bound on s
There is no (53, 68, large)-net in base 9, because
- 13 times m-reduction [i] would yield (53, 55, large)-net in base 9, but