Best Known (126, 126+25, s)-Nets in Base 3
(126, 126+25, 1642)-Net over F3 — Constructive and digital
Digital (126, 151, 1642)-net over F3, using
- 32 times duplication [i] based on digital (124, 149, 1642)-net over F3, using
- net defined by OOA [i] based on linear OOA(3149, 1642, F3, 25, 25) (dual of [(1642, 25), 40901, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
- net defined by OOA [i] based on linear OOA(3149, 1642, F3, 25, 25) (dual of [(1642, 25), 40901, 26]-NRT-code), using
(126, 126+25, 7692)-Net over F3 — Digital
Digital (126, 151, 7692)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3151, 7692, F3, 2, 25) (dual of [(7692, 2), 15233, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3151, 9854, F3, 2, 25) (dual of [(9854, 2), 19557, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3151, 19708, F3, 25) (dual of [19708, 19557, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3127, 19684, F3, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(3151, 19708, F3, 25) (dual of [19708, 19557, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3151, 9854, F3, 2, 25) (dual of [(9854, 2), 19557, 26]-NRT-code), using
(126, 126+25, 2434137)-Net in Base 3 — Upper bound on s
There is no (126, 151, 2434138)-net in base 3, because
- 1 times m-reduction [i] would yield (126, 150, 2434138)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 369989 978603 766503 692358 857191 064844 749479 410501 690410 855812 467646 875769 > 3150 [i]