Best Known (39, 82, s)-Nets in Base 32
(39, 82, 218)-Net over F32 — Constructive and digital
Digital (39, 82, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 28, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (11, 54, 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 (7, 28, 98)-net over F32, using
(39, 82, 288)-Net in Base 32 — Constructive
(39, 82, 288)-net in base 32, using
- 23 times m-reduction [i] based on (39, 105, 288)-net in base 32, using
- base change [i] based on digital (9, 75, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 75, 288)-net over F128, using
(39, 82, 501)-Net over F32 — Digital
Digital (39, 82, 501)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3282, 501, F32, 2, 43) (dual of [(501, 2), 920, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3282, 513, F32, 2, 43) (dual of [(513, 2), 944, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3282, 1026, F32, 43) (dual of [1026, 944, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- linear OA(3282, 1024, F32, 43) (dual of [1024, 942, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3280, 1024, F32, 42) (dual of [1024, 944, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,41], 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
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- OOA 2-folding [i] based on linear OA(3282, 1026, F32, 43) (dual of [1026, 944, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(3282, 513, F32, 2, 43) (dual of [(513, 2), 944, 44]-NRT-code), using
(39, 82, 178930)-Net in Base 32 — Upper bound on s
There is no (39, 82, 178931)-net in base 32, because
- 1 times m-reduction [i] would yield (39, 81, 178931)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 82 640557 608350 775523 832731 642253 700416 479636 511591 667809 389133 758739 477678 211384 750302 391024 320305 836371 814999 071586 054416 > 3281 [i]