Best Known (87, 87+31, s)-Nets in Base 4
(87, 87+31, 531)-Net over F4 — Constructive and digital
Digital (87, 118, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (87, 120, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
(87, 87+31, 1023)-Net over F4 — Digital
Digital (87, 118, 1023)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4118, 1023, F4, 31) (dual of [1023, 905, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4118, 1040, F4, 31) (dual of [1040, 922, 32]-code), using
- construction XX applied to C1 = C([1021,26]), C2 = C([0,28]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1021,28]) [i] based on
- linear OA(4111, 1023, F4, 29) (dual of [1023, 912, 30]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,26}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4106, 1023, F4, 29) (dual of [1023, 917, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4116, 1023, F4, 31) (dual of [1023, 907, 32]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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(41, 6, F4, 1) (dual of [6, 5, 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 (see above)
- construction XX applied to C1 = C([1021,26]), C2 = C([0,28]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1021,28]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4118, 1040, F4, 31) (dual of [1040, 922, 32]-code), using
(87, 87+31, 106332)-Net in Base 4 — Upper bound on s
There is no (87, 118, 106333)-net in base 4, because
- 1 times m-reduction [i] would yield (87, 117, 106333)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 27610 566110 420483 934776 905944 724677 709461 968582 422304 232545 294336 891248 > 4117 [i]