Best Known (77, 77+27, s)-Nets in Base 32
(77, 77+27, 2691)-Net over F32 — Constructive and digital
Digital (77, 104, 2691)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (12, 25, 171)-net over F32, using
- net defined by OOA [i] based on linear OOA(3225, 171, F32, 13, 13) (dual of [(171, 13), 2198, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3225, 1027, F32, 13) (dual of [1027, 1002, 14]-code), using
- construction XX applied to C1 = C([1022,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1022,11]) [i] based on
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3221, 1023, F32, 11) (dual of [1023, 1002, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([1022,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1022,11]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(3225, 1027, F32, 13) (dual of [1027, 1002, 14]-code), using
- net defined by OOA [i] based on linear OOA(3225, 171, F32, 13, 13) (dual of [(171, 13), 2198, 14]-NRT-code), using
- digital (52, 79, 2520)-net over F32, using
- net defined by OOA [i] based on linear OOA(3279, 2520, F32, 27, 27) (dual of [(2520, 27), 67961, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(3279, 32768, F32, 27) (dual of [32768, 32689, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3279, 32761, F32, 27) (dual of [32761, 32682, 28]-code), using
- net defined by OOA [i] based on linear OOA(3279, 2520, F32, 27, 27) (dual of [(2520, 27), 67961, 28]-NRT-code), using
- digital (12, 25, 171)-net over F32, using
(77, 77+27, 20167)-Net in Base 32 — Constructive
(77, 104, 20167)-net in base 32, using
- 321 times duplication [i] based on (76, 103, 20167)-net in base 32, using
- net defined by OOA [i] based on OOA(32103, 20167, S32, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(32103, 262172, S32, 27), using
- 1 times code embedding in larger space [i] based on OA(32102, 262171, S32, 27), using
- discarding parts of the base [i] based on linear OA(6485, 262171, F64, 27) (dual of [262171, 262086, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(6479, 262144, F64, 27) (dual of [262144, 262065, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(646, 27, F64, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(6485, 262171, F64, 27) (dual of [262171, 262086, 28]-code), using
- 1 times code embedding in larger space [i] based on OA(32102, 262171, S32, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(32103, 262172, S32, 27), using
- net defined by OOA [i] based on OOA(32103, 20167, S32, 27, 27), using
(77, 77+27, 356900)-Net over F32 — Digital
Digital (77, 104, 356900)-net over F32, using
(77, 77+27, large)-Net in Base 32 — Upper bound on s
There is no (77, 104, large)-net in base 32, because
- 25 times m-reduction [i] would yield (77, 79, large)-net in base 32, but