Best Known (171−21, 171, s)-Nets in Base 3
(171−21, 171, 53146)-Net over F3 — Constructive and digital
Digital (150, 171, 53146)-net over F3, using
- 31 times duplication [i] based on digital (149, 170, 53146)-net over F3, using
- net defined by OOA [i] based on linear OOA(3170, 53146, F3, 21, 21) (dual of [(53146, 21), 1115896, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3170, 531461, F3, 21) (dual of [531461, 531291, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 531467, F3, 21) (dual of [531467, 531297, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3169, 531442, F3, 21) (dual of [531442, 531273, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 25, F3, 1) (dual of [25, 24, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 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(3170, 531467, F3, 21) (dual of [531467, 531297, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3170, 531461, F3, 21) (dual of [531461, 531291, 22]-code), using
- net defined by OOA [i] based on linear OOA(3170, 53146, F3, 21, 21) (dual of [(53146, 21), 1115896, 22]-NRT-code), using
(171−21, 171, 177156)-Net over F3 — Digital
Digital (150, 171, 177156)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3171, 177156, F3, 3, 21) (dual of [(177156, 3), 531297, 22]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3170, 177156, F3, 3, 21) (dual of [(177156, 3), 531298, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3170, 531468, F3, 21) (dual of [531468, 531298, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3169, 531442, F3, 21) (dual of [531442, 531273, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 3-folding [i] based on linear OA(3170, 531468, F3, 21) (dual of [531468, 531298, 22]-code), using
- 31 times duplication [i] based on linear OOA(3170, 177156, F3, 3, 21) (dual of [(177156, 3), 531298, 22]-NRT-code), using
(171−21, 171, large)-Net in Base 3 — Upper bound on s
There is no (150, 171, large)-net in base 3, because
- 19 times m-reduction [i] would yield (150, 152, large)-net in base 3, but