Best Known (140, 167, s)-Nets in Base 4
(140, 167, 5043)-Net over F4 — Constructive and digital
Digital (140, 167, 5043)-net over F4, using
- 41 times duplication [i] based on digital (139, 166, 5043)-net over F4, using
- net defined by OOA [i] based on linear OOA(4166, 5043, F4, 27, 27) (dual of [(5043, 27), 135995, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4166, 65560, F4, 27) (dual of [65560, 65394, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4166, 65565, F4, 27) (dual of [65565, 65399, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4166, 65565, F4, 27) (dual of [65565, 65399, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4166, 65560, F4, 27) (dual of [65560, 65394, 28]-code), using
- net defined by OOA [i] based on linear OOA(4166, 5043, F4, 27, 27) (dual of [(5043, 27), 135995, 28]-NRT-code), using
(140, 167, 33723)-Net over F4 — Digital
Digital (140, 167, 33723)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4167, 33723, F4, 27) (dual of [33723, 33556, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4167, 65567, F4, 27) (dual of [65567, 65400, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(26) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4167, 65567, F4, 27) (dual of [65567, 65400, 28]-code), using
(140, 167, large)-Net in Base 4 — Upper bound on s
There is no (140, 167, large)-net in base 4, because
- 25 times m-reduction [i] would yield (140, 142, large)-net in base 4, but