Best Known (90, 90+25, s)-Nets in Base 9
(90, 90+25, 4922)-Net over F9 — Constructive and digital
Digital (90, 115, 4922)-net over F9, using
- 91 times duplication [i] based on digital (89, 114, 4922)-net over F9, using
- net defined by OOA [i] based on linear OOA(9114, 4922, F9, 25, 25) (dual of [(4922, 25), 122936, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9114, 59065, F9, 25) (dual of [59065, 58951, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9114, 59067, F9, 25) (dual of [59067, 58953, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(93, 18, F9, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(9114, 59067, F9, 25) (dual of [59067, 58953, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9114, 59065, F9, 25) (dual of [59065, 58951, 26]-code), using
- net defined by OOA [i] based on linear OOA(9114, 4922, F9, 25, 25) (dual of [(4922, 25), 122936, 26]-NRT-code), using
(90, 90+25, 59074)-Net over F9 — Digital
Digital (90, 115, 59074)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9115, 59074, F9, 25) (dual of [59074, 58959, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(9111, 59050, F9, 25) (dual of [59050, 58939, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(991, 59050, F9, 21) (dual of [59050, 58959, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
(90, 90+25, large)-Net in Base 9 — Upper bound on s
There is no (90, 115, large)-net in base 9, because
- 23 times m-reduction [i] would yield (90, 92, large)-net in base 9, but