No, that's a confabulation. The Bible says Jesus was wrapped in a sindon. One of the Gospels, Matthew 27:59, says that Joseph of Arimathea bought a σινδονι (a Sindon)... a fine Linen sheet... not strips. The Shroud is a fine Linen Sheet.
The Gospel of Mark, in Chapter 15 Verse 46 says that Joseph bought a fine (singular) linen cloth, in Greek a σινδονα... again a large sheet.
Only John speaks of "strips" and that may have been referring to the strips or band used to bind the wrists and ankles to keep them from flopping.
"και καθελων αυτο ενετυλιξεν αυτο σινδονι και εθηκεν αυτο εν μνηματι λαξευτω ου ουκ ην ουδεπω ουδεις κειμενος"
You raise an interesting point that a lot of people don't understand about the four books, in that, they describe different things within the same event. They are not discrepancies as some people interpret them, but rather, they are complimentary observations. It's a recurring theme that can be found throughout the Gospels if you read them carefully and diligently, and if you're wise enough, you understand why God chose to do it this way deliberately.
Good thread, and very informative posts, Swordmaker.