Best Known (133−23, 133, s)-Nets in Base 4
(133−23, 133, 1500)-Net over F4 — Constructive and digital
Digital (110, 133, 1500)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (97, 120, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- digital (2, 13, 10)-net over F4, using
(133−23, 133, 16435)-Net over F4 — Digital
Digital (110, 133, 16435)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4133, 16435, F4, 23) (dual of [16435, 16302, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4129, 16428, F4, 23) (dual of [16428, 16299, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(49, 44, F4, 5) (dual of [44, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4129, 16431, F4, 21) (dual of [16431, 16302, 22]-code), using Gilbert–Varšamov bound and bm = 4129 > Vbs−1(k−1) = 2912 084288 117974 597007 097416 850026 879962 820932 812468 445015 688185 357118 952334 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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(4129, 16428, F4, 23) (dual of [16428, 16299, 24]-code), using
- construction X with Varšamov bound [i] based on
(133−23, 133, large)-Net in Base 4 — Upper bound on s
There is no (110, 133, large)-net in base 4, because
- 21 times m-reduction [i] would yield (110, 112, large)-net in base 4, but