Best Known (107−27, 107, s)-Nets in Base 4
(107−27, 107, 531)-Net over F4 — Constructive and digital
Digital (80, 107, 531)-net over F4, using
- t-expansion [i] based on digital (79, 107, 531)-net over F4, using
- 1 times m-reduction [i] based on digital (79, 108, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 177)-net over F64, using
- 1 times m-reduction [i] based on digital (79, 108, 531)-net over F4, using
(107−27, 107, 576)-Net in Base 4 — Constructive
(80, 107, 576)-net in base 4, using
- 42 times duplication [i] based on (78, 105, 576)-net in base 4, using
- trace code for nets [i] based on (8, 35, 192)-net in base 64, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- trace code for nets [i] based on (8, 35, 192)-net in base 64, using
(107−27, 107, 1099)-Net over F4 — Digital
Digital (80, 107, 1099)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4107, 1099, F4, 27) (dual of [1099, 992, 28]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 27 times 0) [i] based on linear OA(4101, 1033, F4, 27) (dual of [1033, 932, 28]-code), using
- construction XX applied to C1 = C([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- linear OA(496, 1023, F4, 26) (dual of [1023, 927, 27]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(496, 1023, F4, 26) (dual of [1023, 927, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [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 = {−1,0,…,25}, and designed minimum distance d ≥ |I|+1 = 28 [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(40, 5, F4, 0) (dual of [5, 5, 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
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1022,25]) [i] based on
- 60 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 27 times 0) [i] based on linear OA(4101, 1033, F4, 27) (dual of [1033, 932, 28]-code), using
(107−27, 107, 153235)-Net in Base 4 — Upper bound on s
There is no (80, 107, 153236)-net in base 4, because
- 1 times m-reduction [i] would yield (80, 106, 153236)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6582 141016 929921 999945 261809 489100 833735 292444 359178 999559 869412 > 4106 [i]