Best Known (94−44, 94, s)-Nets in Base 32
(94−44, 94, 240)-Net over F32 — Constructive and digital
Digital (50, 94, 240)-net over F32, using
- 12 times m-reduction [i] based on digital (50, 106, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 39, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 67, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 39, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(94−44, 94, 513)-Net in Base 32 — Constructive
(50, 94, 513)-net in base 32, using
- t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 14 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(94−44, 94, 1118)-Net over F32 — Digital
Digital (50, 94, 1118)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3294, 1118, F32, 44) (dual of [1118, 1024, 45]-code), using
- 81 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 21 times 0, 1, 39 times 0) [i] based on linear OA(3284, 1027, F32, 44) (dual of [1027, 943, 45]-code), using
- construction XX applied to C1 = C([1022,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([1022,42]) [i] based on
- linear OA(3282, 1023, F32, 43) (dual of [1023, 941, 44]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,41}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3282, 1023, F32, 43) (dual of [1023, 941, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3284, 1023, F32, 44) (dual of [1023, 939, 45]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,42}, and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(3280, 1023, F32, 42) (dual of [1023, 943, 43]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,41], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([1022,42]) [i] based on
- 81 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 21 times 0, 1, 39 times 0) [i] based on linear OA(3284, 1027, F32, 44) (dual of [1027, 943, 45]-code), using
(94−44, 94, 788098)-Net in Base 32 — Upper bound on s
There is no (50, 94, 788099)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 3048 622511 268769 043974 972737 181382 842297 900464 123814 000035 891836 553412 333202 829146 344936 913495 117432 176786 957833 942127 492482 281905 475167 628064 > 3294 [i]