Best Known (126−23, 126, s)-Nets in Base 9
(126−23, 126, 48315)-Net over F9 — Constructive and digital
Digital (103, 126, 48315)-net over F9, using
- 91 times duplication [i] based on digital (102, 125, 48315)-net over F9, using
- net defined by OOA [i] based on linear OOA(9125, 48315, F9, 23, 23) (dual of [(48315, 23), 1111120, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9125, 531466, F9, 23) (dual of [531466, 531341, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9125, 531470, F9, 23) (dual of [531470, 531345, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [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(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-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(9125, 531470, F9, 23) (dual of [531470, 531345, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9125, 531466, F9, 23) (dual of [531466, 531341, 24]-code), using
- net defined by OOA [i] based on linear OOA(9125, 48315, F9, 23, 23) (dual of [(48315, 23), 1111120, 24]-NRT-code), using
(126−23, 126, 519287)-Net over F9 — Digital
Digital (103, 126, 519287)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9126, 519287, F9, 23) (dual of [519287, 519161, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9126, 531471, F9, 23) (dual of [531471, 531345, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9125, 531470, F9, 23) (dual of [531470, 531345, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [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(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(9125, 531470, F9, 23) (dual of [531470, 531345, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9126, 531471, F9, 23) (dual of [531471, 531345, 24]-code), using
(126−23, 126, large)-Net in Base 9 — Upper bound on s
There is no (103, 126, large)-net in base 9, because
- 21 times m-reduction [i] would yield (103, 105, large)-net in base 9, but