Best Known (23−10, 23, s)-Nets in Base 32
(23−10, 23, 207)-Net over F32 — Constructive and digital
Digital (13, 23, 207)-net over F32, using
- 1 times m-reduction [i] based on digital (13, 24, 207)-net over F32, using
- net defined by OOA [i] based on linear OOA(3224, 207, F32, 11, 11) (dual of [(207, 11), 2253, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- 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(3213, 1025, F32, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3224, 1036, F32, 11) (dual of [1036, 1012, 12]-code), using
- net defined by OOA [i] based on linear OOA(3224, 207, F32, 11, 11) (dual of [(207, 11), 2253, 12]-NRT-code), using
(23−10, 23, 819)-Net in Base 32 — Constructive
(13, 23, 819)-net in base 32, using
- net defined by OOA [i] based on OOA(3223, 819, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3223, 4095, S32, 10), using
- discarding factors based on OA(3223, 4098, S32, 10), using
- discarding parts of the base [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
- discarding factors based on OA(3223, 4098, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3223, 4095, S32, 10), using
(23−10, 23, 1131)-Net over F32 — Digital
Digital (13, 23, 1131)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3223, 1131, F32, 10) (dual of [1131, 1108, 11]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 15 times 0, 1, 81 times 0) [i] based on linear OA(3219, 1027, F32, 10) (dual of [1027, 1008, 11]-code), using
- construction XX applied to C1 = C([1022,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([1022,8]) [i] based on
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3219, 1023, F32, 10) (dual of [1023, 1004, 11]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [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([1022,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([1022,8]) [i] based on
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 15 times 0, 1, 81 times 0) [i] based on linear OA(3219, 1027, F32, 10) (dual of [1027, 1008, 11]-code), using
(23−10, 23, 704957)-Net in Base 32 — Upper bound on s
There is no (13, 23, 704958)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 41538 431866 649007 149744 730382 932933 > 3223 [i]