Best Known (191−32, 191, s)-Nets in Base 3
(191−32, 191, 1230)-Net over F3 — Constructive and digital
Digital (159, 191, 1230)-net over F3, using
- 31 times duplication [i] based on digital (158, 190, 1230)-net over F3, using
- net defined by OOA [i] based on linear OOA(3190, 1230, F3, 32, 32) (dual of [(1230, 32), 39170, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3190, 19680, F3, 32) (dual of [19680, 19490, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using
- an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- discarding factors / shortening the dual code based on linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3190, 19680, F3, 32) (dual of [19680, 19490, 33]-code), using
- net defined by OOA [i] based on linear OOA(3190, 1230, F3, 32, 32) (dual of [(1230, 32), 39170, 33]-NRT-code), using
(191−32, 191, 7481)-Net over F3 — Digital
Digital (159, 191, 7481)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3191, 7481, F3, 2, 32) (dual of [(7481, 2), 14771, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3191, 9846, F3, 2, 32) (dual of [(9846, 2), 19501, 33]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3190, 9846, F3, 2, 32) (dual of [(9846, 2), 19502, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3190, 19692, F3, 32) (dual of [19692, 19502, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(3190, 19692, F3, 32) (dual of [19692, 19502, 33]-code), using
- 31 times duplication [i] based on linear OOA(3190, 9846, F3, 2, 32) (dual of [(9846, 2), 19502, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3191, 9846, F3, 2, 32) (dual of [(9846, 2), 19501, 33]-NRT-code), using
(191−32, 191, 1687095)-Net in Base 3 — Upper bound on s
There is no (159, 191, 1687096)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 494627 737477 008848 981135 642759 101311 148435 321783 720469 271119 513802 261664 802347 919331 039233 > 3191 [i]