Mr Fox was gunned down while drinking in the Donegal Celtic Social Club on the Suffolk Road, west Belfast, last Sunday.
More than 100 people were in the venue at the time.
Speaking at the Policing Board yesterday, Mr Byrne said the probe into his murder “remains a fast-paced and live investigation”.
He said the senior investigating officer is pursuing a number of “active lines of enquiry” and “various hypotheses”.
The PSNI has stepped up foot patrols in the area to help reassure the public, Mr Byrne added.
Earlier this week detectives released CCTV images showing the two suspected gunmen entering and leaving the club.
The gunmen fired 20 shots in 21 seconds before fleeing the scene. At one point Mr Fox’s killers stood over him while he was lying on the club’s floor.
It is believed one line of enquiry is the victim’s friendship with Jim Donegan, who was shot dead outside a school in Belfast in December 2018.
Multiple arrests have been made in connection with the Donegan shooting, but no one has been charged.
Speaking to the BBC after yesterday’s board meeting, Mr Byrne said he was keen for public co-operation into the Fox case. He said the shooting was “unusual given the amount of people in attendance”.
“Someone must have some indication about who did what when,” said Mr Byrne.
“My appeal is if you know who has done this — step forward.”
Mr Fox was shot in front of one of his two children and his uncle while he watched a football match. Detective Superintendent Eamonn Corrigan described the killing as “a calculated, planned, ruthless execution”.
He continued “The gunmen arrived, carried out the murder and left the club in under a minute. It was carried out in broad daylight and in the presence of others.”
He added: “The gunmen fired 20 bullets and we have identified multiple strike marks. Anyone in that room or indeed nearby, could have been killed or seriously injured on Sunday
“This cold-blooded act has taken a man’s life. It has left a family bereft, and many others in the local community shocked. This barbaric act has no place in any society.”
Earlier this week, a former senior PSNI officer said Mr Fox’s murder “has the characteristics” of a so-called ‘professional’ hit.
Alan McQuillan, a one time Assistant Chief Constable, said the killers showed a chilling level of organisation.
He said the murder was “a very well-organised incident — they knew exactly where [Mr Fox] would be and what time he would be there, and they were able to walk into the venue with ease and carry out the murder.”
A member of the so-called Marbella Crew of high-profile drug dealers, Mr Fox had been informed he was under threat, but believed himself to be safe because of his longstanding family connections to the social club.
Mr McQuillan added: “The main difference between the average murder and a drug-related murder is that the average murder is local.
“This has the characteristics of a professional hit, which is common in cases involving organised crime.”
Donegal Celtic Sports & Social Club paid tribute to staff following the murder of Mr Fox.
Writing on their Facebook page, the club expressed their “sincere condolences to the Fox family at this very sad time after the passing of our member Sean.”
The club added: “Our thoughts are also with our staff, members and guests who were present at the time.”
The increase in police foot patrols in the area near the shooting to help reassure the public comes as the PSNI is facing questions concerning a ‘recruitment freeze’ which could affect neighbourhood policing.