Best Known (54, 54+16, s)-Nets in Base 32
(54, 54+16, 131116)-Net over F32 — Constructive and digital
Digital (54, 70, 131116)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (45, 61, 131072)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 131072, F32, 16, 16) (dual of [(131072, 16), 2097091, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using
- net defined by OOA [i] based on linear OOA(3261, 131072, F32, 16, 16) (dual of [(131072, 16), 2097091, 17]-NRT-code), using
- digital (1, 9, 44)-net over F32, using
(54, 54+16, 262146)-Net in Base 32 — Constructive
(54, 70, 262146)-net in base 32, using
- base change [i] based on digital (34, 50, 262146)-net over F128, using
- net defined by OOA [i] based on linear OOA(12850, 262146, F128, 16, 16) (dual of [(262146, 16), 4194286, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12850, 2097168, F128, 16) (dual of [2097168, 2097118, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12850, 2097171, F128, 16) (dual of [2097171, 2097121, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(12850, 2097171, F128, 16) (dual of [2097171, 2097121, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12850, 2097168, F128, 16) (dual of [2097168, 2097118, 17]-code), using
- net defined by OOA [i] based on linear OOA(12850, 262146, F128, 16, 16) (dual of [(262146, 16), 4194286, 17]-NRT-code), using
(54, 54+16, 2189978)-Net over F32 — Digital
Digital (54, 70, 2189978)-net over F32, using
(54, 54+16, large)-Net in Base 32 — Upper bound on s
There is no (54, 70, large)-net in base 32, because
- 14 times m-reduction [i] would yield (54, 56, large)-net in base 32, but