Best Known (91−43, 91, s)-Nets in Base 27
(91−43, 91, 196)-Net over F27 — Constructive and digital
Digital (48, 91, 196)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 30, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (18, 61, 108)-net over F27, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- F3 from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F27 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- digital (9, 30, 88)-net over F27, using
(91−43, 91, 370)-Net in Base 27 — Constructive
(48, 91, 370)-net in base 27, using
- t-expansion [i] based on (43, 91, 370)-net in base 27, using
- 17 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- 17 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(91−43, 91, 830)-Net over F27 — Digital
Digital (48, 91, 830)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2791, 830, F27, 43) (dual of [830, 739, 44]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 43 times 0) [i] based on linear OA(2782, 732, F27, 43) (dual of [732, 650, 44]-code), using
- construction XX applied to C1 = C([727,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([727,41]) [i] based on
- linear OA(2780, 728, F27, 42) (dual of [728, 648, 43]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,40}, and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2780, 728, F27, 42) (dual of [728, 648, 43]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,41], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2782, 728, F27, 43) (dual of [728, 646, 44]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,41}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2778, 728, F27, 41) (dual of [728, 650, 42]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,40], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([727,41]) [i] based on
- 89 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 5 times 0, 1, 11 times 0, 1, 24 times 0, 1, 43 times 0) [i] based on linear OA(2782, 732, F27, 43) (dual of [732, 650, 44]-code), using
(91−43, 91, 454909)-Net in Base 27 — Upper bound on s
There is no (48, 91, 454910)-net in base 27, because
- 1 times m-reduction [i] would yield (48, 90, 454910)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 664 887590 089484 701231 667456 671562 850005 160375 934939 319861 435181 811687 219460 359147 652955 491280 614403 420901 841461 529277 137659 521093 > 2790 [i]