That there is no [79,3,71] code follows from Lemma 5.4 of
Simeon Ball, Multiple blocking sets and arcs in finite planes, J. London Math. Soc., 54 (1996) 581--593.
The [78,3,70] code found by Aaron Gulliver is generated by the matrix
01611811415200112911701101313(10)12515712411913414217501213516(10)11601811(10)01(10)12(10)158
601811411215100912711101301(10)13512715412911413214517201513(10)16611801(10)11(10)01(10)12815
1601811415210102911711101303(10)12515712411913414217511203516(10)11611801(10)11(10)02(10)1581
It has weight distribution
WEIGHT (COUNT)
0 (1)
70 (810)
71 (180)
72 (160)
73 (90)
74 (30)
75 (30)
78 (30)
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Updated 28 June 2010