Best Known (100, 123, s)-Nets in Base 9
(100, 123, 48314)-Net over F9 — Constructive and digital
Digital (100, 123, 48314)-net over F9, using
- 91 times duplication [i] based on digital (99, 122, 48314)-net over F9, using
- net defined by OOA [i] based on linear OOA(9122, 48314, F9, 23, 23) (dual of [(48314, 23), 1111100, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(9121, 531442, F9, 23) (dual of [531442, 531321, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(9109, 531442, F9, 21) (dual of [531442, 531333, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 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 C([0,11]) ⊂ C([0,10]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- net defined by OOA [i] based on linear OOA(9122, 48314, F9, 23, 23) (dual of [(48314, 23), 1111100, 24]-NRT-code), using
(100, 123, 379388)-Net over F9 — Digital
Digital (100, 123, 379388)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9123, 379388, F9, 23) (dual of [379388, 379265, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9123, 531456, F9, 23) (dual of [531456, 531333, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(9121, 531442, F9, 23) (dual of [531442, 531321, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(9109, 531442, F9, 21) (dual of [531442, 531333, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 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 C([0,11]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9123, 531456, F9, 23) (dual of [531456, 531333, 24]-code), using
(100, 123, large)-Net in Base 9 — Upper bound on s
There is no (100, 123, large)-net in base 9, because
- 21 times m-reduction [i] would yield (100, 102, large)-net in base 9, but