Best Known (93−43, 93, s)-Nets in Base 32
(93−43, 93, 240)-Net over F32 — Constructive and digital
Digital (50, 93, 240)-net over F32, using
- 13 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
(93−43, 93, 513)-Net in Base 32 — Constructive
(50, 93, 513)-net in base 32, using
- t-expansion [i] based on (46, 93, 513)-net in base 32, using
- 15 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
- 15 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(93−43, 93, 1180)-Net over F32 — Digital
Digital (50, 93, 1180)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3293, 1180, F32, 43) (dual of [1180, 1087, 44]-code), using
- 140 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 20 times 0, 1, 37 times 0, 1, 63 times 0) [i] based on linear OA(3283, 1030, F32, 43) (dual of [1030, 947, 44]-code), using
- construction XX applied to C1 = C([1021,39]), C2 = C([0,40]), C3 = C1 + C2 = C([0,39]), and C∩ = C1 ∩ C2 = C([1021,40]) [i] based on
- linear OA(3280, 1023, F32, 42) (dual of [1023, 943, 43]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,39}, and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,40], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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 = {−2,−1,…,40}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3276, 1023, F32, 40) (dual of [1023, 947, 41]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,39], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- 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
- construction XX applied to C1 = C([1021,39]), C2 = C([0,40]), C3 = C1 + C2 = C([0,39]), and C∩ = C1 ∩ C2 = C([1021,40]) [i] based on
- 140 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 20 times 0, 1, 37 times 0, 1, 63 times 0) [i] based on linear OA(3283, 1030, F32, 43) (dual of [1030, 947, 44]-code), using
(93−43, 93, 1099302)-Net in Base 32 — Upper bound on s
There is no (50, 93, 1099303)-net in base 32, because
- 1 times m-reduction [i] would yield (50, 92, 1099303)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 2 977185 657399 749917 128057 629987 384177 844895 500883 896730 881684 516398 692862 077058 533648 272913 331931 790404 294412 533677 458879 405932 660947 696044 > 3292 [i]