Best Known (140−23, 140, s)-Nets in Base 4
(140−23, 140, 5959)-Net over F4 — Constructive and digital
Digital (117, 140, 5959)-net over F4, using
- 42 times duplication [i] based on digital (115, 138, 5959)-net over F4, using
- net defined by OOA [i] based on linear OOA(4138, 5959, F4, 23, 23) (dual of [(5959, 23), 136919, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4138, 65550, F4, 23) (dual of [65550, 65412, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4138, 65553, F4, 23) (dual of [65553, 65415, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4138, 65553, F4, 23) (dual of [65553, 65415, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4138, 65550, F4, 23) (dual of [65550, 65412, 24]-code), using
- net defined by OOA [i] based on linear OOA(4138, 5959, F4, 23, 23) (dual of [(5959, 23), 136919, 24]-NRT-code), using
(140−23, 140, 32778)-Net over F4 — Digital
Digital (117, 140, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4140, 32778, F4, 2, 23) (dual of [(32778, 2), 65416, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4140, 65556, F4, 23) (dual of [65556, 65416, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- 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(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(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(22) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(4140, 65556, F4, 23) (dual of [65556, 65416, 24]-code), using
(140−23, 140, large)-Net in Base 4 — Upper bound on s
There is no (117, 140, large)-net in base 4, because
- 21 times m-reduction [i] would yield (117, 119, large)-net in base 4, but