Best Known (92−43, 92, s)-Nets in Base 32
(92−43, 92, 240)-Net over F32 — Constructive and digital
Digital (49, 92, 240)-net over F32, using
- 11 times m-reduction [i] based on digital (49, 103, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 38, 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, 65, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 38, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(92−43, 92, 513)-Net in Base 32 — Constructive
(49, 92, 513)-net in base 32, using
- t-expansion [i] based on (46, 92, 513)-net in base 32, using
- 16 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
- 16 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(92−43, 92, 1115)-Net over F32 — Digital
Digital (49, 92, 1115)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3292, 1115, F32, 43) (dual of [1115, 1023, 44]-code), using
- 78 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 0, 0, 0, 1, 10 times 0, 1, 20 times 0, 1, 38 times 0) [i] based on linear OA(3282, 1027, F32, 43) (dual of [1027, 945, 44]-code), using
- construction XX applied to C1 = C([1022,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1022,41]) [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 = {−1,0,…,40}, and designed minimum distance d ≥ |I|+1 = 43 [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(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(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(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([1022,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1022,41]) [i] based on
- 78 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 0, 0, 0, 1, 10 times 0, 1, 20 times 0, 1, 38 times 0) [i] based on linear OA(3282, 1027, F32, 43) (dual of [1027, 945, 44]-code), using
(92−43, 92, 932056)-Net in Base 32 — Upper bound on s
There is no (49, 92, 932057)-net in base 32, because
- 1 times m-reduction [i] would yield (49, 91, 932057)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 93035 363443 343430 338361 940983 753892 506733 561462 873716 119714 144875 048288 774887 323062 356169 853387 552679 882911 035172 950499 162933 958528 163064 > 3291 [i]