Best Known (151, 151+22, s)-Nets in Base 4
(151, 151+22, 95335)-Net over F4 — Constructive and digital
Digital (151, 173, 95335)-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 (139, 161, 95326)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
- digital (1, 12, 9)-net over F4, using
(151, 151+22, 524315)-Net over F4 — Digital
Digital (151, 173, 524315)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4173, 524315, F4, 2, 22) (dual of [(524315, 2), 1048457, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4173, 1048630, F4, 22) (dual of [1048630, 1048457, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4169, 1048624, F4, 22) (dual of [1048624, 1048455, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(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(21) ⊂ Ce(16) [i] based on
- linear OA(4169, 1048626, F4, 19) (dual of [1048626, 1048457, 20]-code), using Gilbert–Varšamov bound and bm = 4169 > Vbs−1(k−1) = 142214 676168 425687 962382 179001 005364 558602 026249 348256 279316 539898 592853 537950 928498 269324 977246 760376 [i]
- linear OA(42, 4, F4, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,4)), using
- Reed–Solomon code RS(2,4) [i]
- linear OA(4169, 1048624, F4, 22) (dual of [1048624, 1048455, 23]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4173, 1048630, F4, 22) (dual of [1048630, 1048457, 23]-code), using
(151, 151+22, large)-Net in Base 4 — Upper bound on s
There is no (151, 173, large)-net in base 4, because
- 20 times m-reduction [i] would yield (151, 153, large)-net in base 4, but