Best Known (167, 167+22, s)-Nets in Base 4
(167, 167+22, 381310)-Net over F4 — Constructive and digital
Digital (167, 189, 381310)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (155, 177, 381301)-net over F4, using
- net defined by OOA [i] based on linear OOA(4177, 381301, F4, 22, 22) (dual of [(381301, 22), 8388445, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4177, 4194311, F4, 22) (dual of [4194311, 4194134, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 4194315, F4, 22) (dual of [4194315, 4194138, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4177, 4194315, F4, 22) (dual of [4194315, 4194138, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4177, 4194311, F4, 22) (dual of [4194311, 4194134, 23]-code), using
- net defined by OOA [i] based on linear OOA(4177, 381301, F4, 22, 22) (dual of [(381301, 22), 8388445, 23]-NRT-code), using
- digital (1, 12, 9)-net over F4, using
(167, 167+22, 2097180)-Net over F4 — Digital
Digital (167, 189, 2097180)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4189, 2097180, F4, 2, 22) (dual of [(2097180, 2), 4194171, 23]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4185, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194171, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(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(21) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4185, 4194356, F4, 22) (dual of [4194356, 4194171, 23]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4185, 2097178, F4, 2, 22) (dual of [(2097178, 2), 4194171, 23]-NRT-code), using
(167, 167+22, large)-Net in Base 4 — Upper bound on s
There is no (167, 189, large)-net in base 4, because
- 20 times m-reduction [i] would yield (167, 169, large)-net in base 4, but