The schools admissions watchdog says parents should appeal if they are "disappointed" by the secondary place they have been allocated.
Ian Craig, chief adjudicator for schools in England, spoke as over half a million families found whether they had their first preference place.
Researchers have warned that the admissions process is too complex for many parents.
Last year, about one in five pupils did not get their first choice school.
Schools Secretary Ed Balls has admitted that the system is not perfect but said it is fairer than it had ever been.
Mr Balls said: "We've already given extra powers to the [chief schools] adjudicator, he can now do investigations.
"Any changes in admissions arrangements have to be consulted on with parents."
Ed Balls: 'Everyone should have the chance to go to a good local school'
Mr Balls has asked the adjudicator to look at how many authorities are using a lottery - or "random allocation" - and whether it is fair to children.
He said: "I think in the last resort, lotteries can be the right way when you've got a genuine tie and there's no other way to sort it out.
"But a widespread use of lotteries seems to me pretty arbitrary and unfair."
The research, carried out for the Research and Information on State Education charity, claims that parents are still unclear about the workings of the school admissions system, following efforts by the government to make it more transparent.
"Despite improvements, our research suggests that the system is still too complex, particularly for parents and carers who are not highly educated or proficient in English, and especially where there are schools responsible for their own admissions," says report author, Anne West.
This year has seen the introduction of an updated admissions code, which has attempted to make the process fairer and more open.
With the sending out of the admissions decisions on Monday, attention has been focused on families trying to find a suitable place for their children.
Surveys of parents have shown uncertainty about the success of the current admissions system.
An ICM poll for the Guardian showed that 44% of parents did not believe that school admissions gave children a fair chance - with the better-off parents mostly likely to be sceptical.
While the LSE research pointed to a lack of transparency around admissions to faith schools, the ICM poll found 60% of parents thought pupils benefited from faith schools.
The Advisory Centre for Education (ACE), which offers advice on school issues, says the arrival of the news about places is likely to trigger a wave of inquiries from anxious parents - and it is providing information online.
Last year, the advisory centre found that 40% of its calls between January and March were about admissions and the appeals process.
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