Best Known (124, 124+18, s)-Nets in Base 4
(124, 124+18, 116519)-Net over F4 — Constructive and digital
Digital (124, 142, 116519)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 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 (113, 131, 116509)-net over F4, using
- net defined by OOA [i] based on linear OOA(4131, 116509, F4, 18, 18) (dual of [(116509, 18), 2097031, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4131, 1048581, F4, 18) (dual of [1048581, 1048450, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4131, 1048581, F4, 18) (dual of [1048581, 1048450, 19]-code), using
- net defined by OOA [i] based on linear OOA(4131, 116509, F4, 18, 18) (dual of [(116509, 18), 2097031, 19]-NRT-code), using
- digital (2, 11, 10)-net over F4, using
(124, 124+18, 524314)-Net over F4 — Digital
Digital (124, 142, 524314)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4142, 524314, F4, 2, 18) (dual of [(524314, 2), 1048486, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4142, 1048628, F4, 18) (dual of [1048628, 1048486, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4139, 1048624, F4, 18) (dual of [1048624, 1048485, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4139, 1048625, F4, 15) (dual of [1048625, 1048486, 16]-code), using Gilbert–Varšamov bound and bm = 4139 > Vbs−1(k−1) = 106 642560 461792 409923 627689 380487 584874 430228 575430 777377 059074 615722 713981 444427 [i]
- linear OA(42, 3, F4, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,4)), using
- dual of repetition code with length 3 [i]
- linear OA(4139, 1048624, F4, 18) (dual of [1048624, 1048485, 19]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4142, 1048628, F4, 18) (dual of [1048628, 1048486, 19]-code), using
(124, 124+18, large)-Net in Base 4 — Upper bound on s
There is no (124, 142, large)-net in base 4, because
- 16 times m-reduction [i] would yield (124, 126, large)-net in base 4, but