Best Known (127−21, 127, s)-Nets in Base 3
(127−21, 127, 1969)-Net over F3 — Constructive and digital
Digital (106, 127, 1969)-net over F3, using
- net defined by OOA [i] based on linear OOA(3127, 1969, F3, 21, 21) (dual of [(1969, 21), 41222, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3127, 19691, F3, 21) (dual of [19691, 19564, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 19692, F3, 21) (dual of [19692, 19565, 22]-code), using
- 1 times truncation [i] based on linear OA(3128, 19693, F3, 22) (dual of [19693, 19565, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- 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(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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
- 1 times truncation [i] based on linear OA(3128, 19693, F3, 22) (dual of [19693, 19565, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 19692, F3, 21) (dual of [19692, 19565, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3127, 19691, F3, 21) (dual of [19691, 19564, 22]-code), using
(127−21, 127, 7753)-Net over F3 — Digital
Digital (106, 127, 7753)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 7753, F3, 2, 21) (dual of [(7753, 2), 15379, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3127, 9846, F3, 2, 21) (dual of [(9846, 2), 19565, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3127, 19692, F3, 21) (dual of [19692, 19565, 22]-code), using
- 1 times truncation [i] based on linear OA(3128, 19693, F3, 22) (dual of [19693, 19565, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- 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(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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
- 1 times truncation [i] based on linear OA(3128, 19693, F3, 22) (dual of [19693, 19565, 23]-code), using
- OOA 2-folding [i] based on linear OA(3127, 19692, F3, 21) (dual of [19692, 19565, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(3127, 9846, F3, 2, 21) (dual of [(9846, 2), 19565, 22]-NRT-code), using
(127−21, 127, 2326335)-Net in Base 3 — Upper bound on s
There is no (106, 127, 2326336)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 126, 2326336)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 310023 460383 834269 490879 338630 119251 760653 516170 939419 213697 > 3126 [i]