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The threat letter was sent as a postcard. The card stated bombs would be planted in three venues of the state government's public outreach programme.
It was revealed that 14 of 24 suicide bombings in the country this year were carried out by Afghan nationals.
No group has claimed responsibility for the attack, but previous such attacks have been blamed on Baloch insurgents and Islamic militants.
The bombing was the deadliest in a decade to hit Peshawar.
The attack, in the Police Lines district, was the deadliest in a decade to hit the northwestern city near the Afghan border and comes amid a surge in violence against the police. No group has claimed responsibility.
Turnout in the elections is set to be low, with experts citing fear of violence and a loss of hope among voters following widespread fraud allegations during the 2014 election.