Best Known (125, 125+21, s)-Nets in Base 3
(125, 125+21, 5907)-Net over F3 — Constructive and digital
Digital (125, 146, 5907)-net over F3, using
- 34 times duplication [i] based on digital (121, 142, 5907)-net over F3, using
- net defined by OOA [i] based on linear OOA(3142, 5907, F3, 21, 21) (dual of [(5907, 21), 123905, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3142, 59071, F3, 21) (dual of [59071, 58929, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3141, 59050, F3, 21) (dual of [59050, 58909, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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
- OOA 10-folding and stacking with additional row [i] based on linear OA(3142, 59071, F3, 21) (dual of [59071, 58929, 22]-code), using
- net defined by OOA [i] based on linear OOA(3142, 5907, F3, 21, 21) (dual of [(5907, 21), 123905, 22]-NRT-code), using
(125, 125+21, 24761)-Net over F3 — Digital
Digital (125, 146, 24761)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 24761, F3, 2, 21) (dual of [(24761, 2), 49376, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 29538, F3, 2, 21) (dual of [(29538, 2), 58930, 22]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(3142, 29536, F3, 2, 21) (dual of [(29536, 2), 58930, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3142, 59072, F3, 21) (dual of [59072, 58930, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3141, 59050, F3, 21) (dual of [59050, 58909, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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 2-folding [i] based on linear OA(3142, 59072, F3, 21) (dual of [59072, 58930, 22]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(3142, 29536, F3, 2, 21) (dual of [(29536, 2), 58930, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 29538, F3, 2, 21) (dual of [(29538, 2), 58930, 22]-NRT-code), using
(125, 125+21, large)-Net in Base 3 — Upper bound on s
There is no (125, 146, large)-net in base 3, because
- 19 times m-reduction [i] would yield (125, 127, large)-net in base 3, but