Best Known (149, 149+22, s)-Nets in Base 3
(149, 149+22, 48314)-Net over F3 — Constructive and digital
Digital (149, 171, 48314)-net over F3, using
- 31 times duplication [i] based on digital (148, 170, 48314)-net over F3, using
- net defined by OOA [i] based on linear OOA(3170, 48314, F3, 22, 22) (dual of [(48314, 22), 1062738, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3170, 531454, F3, 22) (dual of [531454, 531284, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 Ce(21) ⊂ Ce(19) [i] based on
- OA 11-folding and stacking [i] based on linear OA(3170, 531454, F3, 22) (dual of [531454, 531284, 23]-code), using
- net defined by OOA [i] based on linear OOA(3170, 48314, F3, 22, 22) (dual of [(48314, 22), 1062738, 23]-NRT-code), using
(149, 149+22, 132863)-Net over F3 — Digital
Digital (149, 171, 132863)-net over F3, using
- 31 times duplication [i] based on digital (148, 170, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3170, 132863, F3, 4, 22) (dual of [(132863, 4), 531282, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3170, 531452, F3, 22) (dual of [531452, 531282, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 531454, F3, 22) (dual of [531454, 531284, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 531454, F3, 22) (dual of [531454, 531284, 23]-code), using
- OOA 4-folding [i] based on linear OA(3170, 531452, F3, 22) (dual of [531452, 531282, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3170, 132863, F3, 4, 22) (dual of [(132863, 4), 531282, 23]-NRT-code), using
(149, 149+22, large)-Net in Base 3 — Upper bound on s
There is no (149, 171, large)-net in base 3, because
- 20 times m-reduction [i] would yield (149, 151, large)-net in base 3, but