Best Known (90−42, 90, s)-Nets in Base 32
(90−42, 90, 240)-Net over F32 — Constructive and digital
Digital (48, 90, 240)-net over F32, using
- 10 times m-reduction [i] based on digital (48, 100, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 37, 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, 63, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 37, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(90−42, 90, 513)-Net in Base 32 — Constructive
(48, 90, 513)-net in base 32, using
- t-expansion [i] based on (46, 90, 513)-net in base 32, using
- 18 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
- 18 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(90−42, 90, 1113)-Net over F32 — Digital
Digital (48, 90, 1113)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3290, 1113, F32, 42) (dual of [1113, 1023, 43]-code), using
- 74 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0) [i] based on linear OA(3281, 1030, F32, 42) (dual of [1030, 949, 43]-code), using
- construction XX applied to C1 = C([1021,38]), C2 = C([0,39]), C3 = C1 + C2 = C([0,38]), and C∩ = C1 ∩ C2 = C([1021,39]) [i] based on
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,38}, and designed minimum distance d ≥ |I|+1 = 42 [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(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(3274, 1023, F32, 39) (dual of [1023, 949, 40]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,38], and designed minimum distance d ≥ |I|+1 = 40 [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([1021,38]), C2 = C([0,39]), C3 = C1 + C2 = C([0,38]), and C∩ = C1 ∩ C2 = C([1021,39]) [i] based on
- 74 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0) [i] based on linear OA(3281, 1030, F32, 42) (dual of [1030, 949, 43]-code), using
(90−42, 90, 790255)-Net in Base 32 — Upper bound on s
There is no (48, 90, 790256)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2907 359413 472739 715742 943173 189152 789794 790246 614495 869953 379656 320357 750710 093214 383983 968312 092839 026898 241438 655644 997819 900503 346686 > 3290 [i]