Best Known (26−13, 26, s)-Nets in Base 32
(26−13, 26, 171)-Net over F32 — Constructive and digital
Digital (13, 26, 171)-net over F32, using
- 321 times duplication [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
(26−13, 26, 259)-Net in Base 32 — Constructive
(13, 26, 259)-net in base 32, using
- 2 times m-reduction [i] based on (13, 28, 259)-net in base 32, using
- base change [i] based on (5, 20, 259)-net in base 128, using
- 4 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
- base change [i] based on digital (2, 21, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 21, 259)-net over F256, using
- 4 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
- base change [i] based on (5, 20, 259)-net in base 128, using
(26−13, 26, 515)-Net over F32 — Digital
Digital (13, 26, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3226, 515, F32, 2, 13) (dual of [(515, 2), 1004, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3226, 1030, F32, 13) (dual of [1030, 1004, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(3225, 1025, F32, 13) (dual of [1025, 1000, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3221, 1025, F32, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(3226, 1030, F32, 13) (dual of [1030, 1004, 14]-code), using
(26−13, 26, 180431)-Net in Base 32 — Upper bound on s
There is no (13, 26, 180432)-net in base 32, because
- 1 times m-reduction [i] would yield (13, 25, 180432)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 42 536127 042223 814882 208168 409718 520349 > 3225 [i]