MinT - Best Known (25, 25+23, s)-Nets in Base 64

Best Known (25, 25+23, s)-Nets in Base 64

(25, 25+23, 373)-Net over F₆₄ — Constructive and digital

Digital (25, 48, 373)-net over F₆₄, using

64¹ times duplication [i] based on digital (24, 47, 373)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁴⁷, 373, F₆₄, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(64⁴⁷, 4104, F₆₄, 23) (dual of [4104, 4057, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
      1. linear OA(64⁴⁵, 4096, F₆₄, 23) (dual of [4096, 4051, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(64³⁹, 4096, F₆₄, 20) (dual of [4096, 4057, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(25, 25+23, 516)-Net in Base 64 — Constructive

(25, 48, 516)-net in base 64, using

base change [i] based on digital (13, 36, 516)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 12, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (1, 24, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(25, 25+23, 1871)-Net over F₆₄ — Digital

Digital (25, 48, 1871)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴⁸, 1871, F₆₄, 2, 23) (dual of [(1871, 2), 3694, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁴⁸, 2054, F₆₄, 2, 23) (dual of [(2054, 2), 4060, 24]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁴⁸, 4108, F₆₄, 23) (dual of [4108, 4060, 24]-code), using
    - constructionÂ X applied to C([0,11]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(64⁴⁵, 4097, F₆₄, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
      2. linear OA(64³⁷, 4097, F₆₄, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(64³, 11, F₆₄, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
        discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(25, 25+23, 4064316)-Net in Base 64 — Upper bound on s

There is no (25, 48, 4064317)-net in base 64, because

1 times m-reduction [i] would yield (25, 47, 4064317)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7 770692 433785 365238 503305 493012 640661 585907 137766 388582 877393 495148 850186 377589 617420 > 64⁴⁷ [i]