Best Known (85, 85+23, s)-Nets in Base 9
(85, 85+23, 5370)-Net over F9 — Constructive and digital
Digital (85, 108, 5370)-net over F9, using
- 93 times duplication [i] based on digital (82, 105, 5370)-net over F9, using
- net defined by OOA [i] based on linear OOA(9105, 5370, F9, 23, 23) (dual of [(5370, 23), 123405, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9105, 59071, F9, 23) (dual of [59071, 58966, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9105, 59074, F9, 23) (dual of [59074, 58969, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(9101, 59050, F9, 23) (dual of [59050, 58949, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(981, 59050, F9, 19) (dual of [59050, 58969, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9105, 59074, F9, 23) (dual of [59074, 58969, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9105, 59071, F9, 23) (dual of [59071, 58966, 24]-code), using
- net defined by OOA [i] based on linear OOA(9105, 5370, F9, 23, 23) (dual of [(5370, 23), 123405, 24]-NRT-code), using
(85, 85+23, 59081)-Net over F9 — Digital
Digital (85, 108, 59081)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9108, 59081, F9, 23) (dual of [59081, 58973, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- 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(976, 59049, F9, 17) (dual of [59049, 58973, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
(85, 85+23, large)-Net in Base 9 — Upper bound on s
There is no (85, 108, large)-net in base 9, because
- 21 times m-reduction [i] would yield (85, 87, large)-net in base 9, but