Best Known (136, 154, s)-Nets in Base 4
(136, 154, 466044)-Net over F4 — Constructive and digital
Digital (136, 154, 466044)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (126, 144, 466035)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 466035, F4, 18, 18) (dual of [(466035, 18), 8388486, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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
- OA 9-folding and stacking [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- net defined by OOA [i] based on linear OOA(4144, 466035, F4, 18, 18) (dual of [(466035, 18), 8388486, 19]-NRT-code), using
- digital (1, 10, 9)-net over F4, using
(136, 154, 2097179)-Net over F4 — Digital
Digital (136, 154, 2097179)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4154, 2097179, F4, 2, 18) (dual of [(2097179, 2), 4194204, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4152, 2097178, F4, 2, 18) (dual of [(2097178, 2), 4194204, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4152, 4194356, F4, 18) (dual of [4194356, 4194204, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(48, 52, F4, 4) (dual of [52, 44, 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
- OOA 2-folding [i] based on linear OA(4152, 4194356, F4, 18) (dual of [4194356, 4194204, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4152, 2097178, F4, 2, 18) (dual of [(2097178, 2), 4194204, 19]-NRT-code), using
(136, 154, large)-Net in Base 4 — Upper bound on s
There is no (136, 154, large)-net in base 4, because
- 16 times m-reduction [i] would yield (136, 138, large)-net in base 4, but