Best Known (127, 127+19, s)-Nets in Base 3
(127, 127+19, 59050)-Net over F3 — Constructive and digital
Digital (127, 146, 59050)-net over F3, using
- net defined by OOA [i] based on linear OOA(3146, 59050, F3, 19, 19) (dual of [(59050, 19), 1121804, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3146, 531451, F3, 19) (dual of [531451, 531305, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3146, 531454, F3, 19) (dual of [531454, 531308, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3146, 531454, F3, 19) (dual of [531454, 531308, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3146, 531451, F3, 19) (dual of [531451, 531305, 20]-code), using
(127, 127+19, 132863)-Net over F3 — Digital
Digital (127, 146, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 132863, F3, 4, 19) (dual of [(132863, 4), 531306, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3146, 531452, F3, 19) (dual of [531452, 531306, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3146, 531454, F3, 19) (dual of [531454, 531308, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3146, 531454, F3, 19) (dual of [531454, 531308, 20]-code), using
- OOA 4-folding [i] based on linear OA(3146, 531452, F3, 19) (dual of [531452, 531306, 20]-code), using
(127, 127+19, large)-Net in Base 3 — Upper bound on s
There is no (127, 146, large)-net in base 3, because
- 17 times m-reduction [i] would yield (127, 129, large)-net in base 3, but