Best Known (111, 133, s)-Nets in Base 4
(111, 133, 5959)-Net over F4 — Constructive and digital
Digital (111, 133, 5959)-net over F4, using
- 41 times duplication [i] based on digital (110, 132, 5959)-net over F4, using
- net defined by OOA [i] based on linear OOA(4132, 5959, F4, 22, 22) (dual of [(5959, 22), 130966, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4132, 65549, F4, 22) (dual of [65549, 65417, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 65555, F4, 22) (dual of [65555, 65423, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- 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(4113, 65536, F4, 19) (dual of [65536, 65423, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4132, 65555, F4, 22) (dual of [65555, 65423, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4132, 65549, F4, 22) (dual of [65549, 65417, 23]-code), using
- net defined by OOA [i] based on linear OOA(4132, 5959, F4, 22, 22) (dual of [(5959, 22), 130966, 23]-NRT-code), using
(111, 133, 32778)-Net over F4 — Digital
Digital (111, 133, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4133, 32778, F4, 2, 22) (dual of [(32778, 2), 65423, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4133, 65556, F4, 22) (dual of [65556, 65423, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, 65557, F4, 22) (dual of [65557, 65424, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- 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(4113, 65536, F4, 19) (dual of [65536, 65423, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), 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(21) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4133, 65557, F4, 22) (dual of [65557, 65424, 23]-code), using
- OOA 2-folding [i] based on linear OA(4133, 65556, F4, 22) (dual of [65556, 65423, 23]-code), using
(111, 133, large)-Net in Base 4 — Upper bound on s
There is no (111, 133, large)-net in base 4, because
- 20 times m-reduction [i] would yield (111, 113, large)-net in base 4, but