Best Known (76, 103, s)-Nets in Base 4
(76, 103, 531)-Net over F4 — Constructive and digital
Digital (76, 103, 531)-net over F4, using
- 41 times duplication [i] based on digital (75, 102, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 177)-net over F64, using
(76, 103, 952)-Net over F4 — Digital
Digital (76, 103, 952)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 952, F4, 27) (dual of [952, 849, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4103, 1040, F4, 27) (dual of [1040, 937, 28]-code), using
- construction XX applied to C1 = C([1021,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1021,24]) [i] based on
- linear OA(496, 1023, F4, 25) (dual of [1023, 927, 26]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,22}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(491, 1023, F4, 25) (dual of [1023, 932, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,24}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(486, 1023, F4, 23) (dual of [1023, 937, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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([1021,22]), C2 = C([0,24]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1021,24]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4103, 1040, F4, 27) (dual of [1040, 937, 28]-code), using
(76, 103, 100021)-Net in Base 4 — Upper bound on s
There is no (76, 103, 100022)-net in base 4, because
- 1 times m-reduction [i] would yield (76, 102, 100022)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 25 711129 801060 826782 475771 604648 312868 014529 795742 158499 110138 > 4102 [i]