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The Chihuahua state prosecutor said in a statement that among those who died in the prison attack were 10 security personnel and four inmates.
'I am responsible for this disappointment. It is a great sadness and I assume all the responsibility of this huge failure,' said Martino, whose contract was reportedly due to expire after Mexico's World Cup campaign.
Mexico finished with four points, level with Poland who lost 2-0 to group winners Argentina and had a superior goal difference of one to advance.
The Argentina team did not immediately respond to a request for comment about Alvarez's statement.
After two rounds, Mexico are bottom of Group C with just one point.
Poland lead the group with four points from two matches followed by Argentina and Saudi (three).
The South Americans have won both their previous encounters in the World Cups – in 2006 and 2010.
Robert Lewandowski's 58th minute penalty was only the second shot on target Poland had all game.
The French woman officiated the Group C match between Mexico and Poland.
Jesus "Tecatito" Corona, who is a key player for Mexico, is currently injured.