Best Known (110−21, 110, s)-Nets in Base 9
(110−21, 110, 53145)-Net over F9 — Constructive and digital
Digital (89, 110, 53145)-net over F9, using
- net defined by OOA [i] based on linear OOA(9110, 53145, F9, 21, 21) (dual of [(53145, 21), 1115935, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9110, 531451, F9, 21) (dual of [531451, 531341, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 531455, F9, 21) (dual of [531455, 531345, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 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(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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9110, 531455, F9, 21) (dual of [531455, 531345, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9110, 531451, F9, 21) (dual of [531451, 531341, 22]-code), using
(110−21, 110, 295426)-Net over F9 — Digital
Digital (89, 110, 295426)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9110, 295426, F9, 21) (dual of [295426, 295316, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9110, 531455, F9, 21) (dual of [531455, 531345, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 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(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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9110, 531455, F9, 21) (dual of [531455, 531345, 22]-code), using
(110−21, 110, large)-Net in Base 9 — Upper bound on s
There is no (89, 110, large)-net in base 9, because
- 19 times m-reduction [i] would yield (89, 91, large)-net in base 9, but