Best Known (174−21, 174, s)-Nets in Base 3
(174−21, 174, 53147)-Net over F3 — Constructive and digital
Digital (153, 174, 53147)-net over F3, using
- 31 times duplication [i] based on digital (152, 173, 53147)-net over F3, using
- net defined by OOA [i] based on linear OOA(3173, 53147, F3, 21, 21) (dual of [(53147, 21), 1115914, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3173, 531471, F3, 21) (dual of [531471, 531298, 22]-code), using
- 3 times code embedding in larger space [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
- 3 times code embedding in larger space [i] based on linear OA(3170, 531468, F3, 21) (dual of [531468, 531298, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3173, 531471, F3, 21) (dual of [531471, 531298, 22]-code), using
- net defined by OOA [i] based on linear OOA(3173, 53147, F3, 21, 21) (dual of [(53147, 21), 1115914, 22]-NRT-code), using
(174−21, 174, 177157)-Net over F3 — Digital
Digital (153, 174, 177157)-net over F3, using
- 31 times duplication [i] based on digital (152, 173, 177157)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3173, 177157, F3, 3, 21) (dual of [(177157, 3), 531298, 22]-NRT-code), using
- 1 times NRT-code embedding in larger space [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
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3170, 177156, F3, 3, 21) (dual of [(177156, 3), 531298, 22]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3173, 177157, F3, 3, 21) (dual of [(177157, 3), 531298, 22]-NRT-code), using
(174−21, 174, large)-Net in Base 3 — Upper bound on s
There is no (153, 174, large)-net in base 3, because
- 19 times m-reduction [i] would yield (153, 155, large)-net in base 3, but