Best Known (157, 157+31, s)-Nets in Base 4
(157, 157+31, 4370)-Net over F4 — Constructive and digital
Digital (157, 188, 4370)-net over F4, using
- 42 times duplication [i] based on digital (155, 186, 4370)-net over F4, using
- net defined by OOA [i] based on linear OOA(4186, 4370, F4, 31, 31) (dual of [(4370, 31), 135284, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4186, 65551, F4, 31) (dual of [65551, 65365, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4186, 65551, F4, 31) (dual of [65551, 65365, 32]-code), using
- net defined by OOA [i] based on linear OOA(4186, 4370, F4, 31, 31) (dual of [(4370, 31), 135284, 32]-NRT-code), using
(157, 157+31, 32778)-Net over F4 — Digital
Digital (157, 188, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4188, 32778, F4, 2, 31) (dual of [(32778, 2), 65368, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4188, 65556, F4, 31) (dual of [65556, 65368, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(41, 18, F4, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(4188, 65556, F4, 31) (dual of [65556, 65368, 32]-code), using
(157, 157+31, large)-Net in Base 4 — Upper bound on s
There is no (157, 188, large)-net in base 4, because
- 29 times m-reduction [i] would yield (157, 159, large)-net in base 4, but