Publishedduration1 hour agoimage copyrightGetty Imagesimage captionAdelaide went into lockdown with the rest of South Australia on Wednesday

South Australia decided to enter a state-wide lockdown after learning a man with Covid-19 had lied about his link to a pizza shop, police say.

The strict six-day lockdown began on Wednesday after the state detected 36 infections, including its first locally acquired cases since April.

But the order would have been avoided if a man had told health officials he had worked "several" shifts at a pizza shop, authorities said.

He had said he was a customer there.

"To say I am fuming is an understatement," state Premier Steven Marshall told reporters on Friday.

Australia has relied on lockdowns, widespread testing and aggressive contact tracing to push daily infections close to zero.

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"We are absolutely livid with the actions of this individual and we will be looking very carefully at what consequences there [are] going to be," Mr Marshall added.

But that would not include police action against the man because there was "no penalty associated with telling lies", said South Australia Police Commissioner Grant Stevens.

"This person's failure to be truthful with contact tracers means his close associates – who we understand he's had close contact with – haven't been identified and are currently in the community," Mr Stevens said.