Are there examples of the shroud weave (linen) from the 1st century to show they had the technology to produce such a weave?
Yes, during the first century they had the technology to produce the herringbone weave.
Various herringbone weaves have been found in antiquity:
A pair of woolen leggings found in the permafrost of the Italian-Austrian Alps have a 2:2 herringbone weave, dating to 800 to 500 BC. A dark blue cloth with a 2:2 herringbone weave was found at Murabba’at Cave in Israel, from the Roman period. A textile with a 2:2 herringbone weave was found at Pompeii, from 79 AD. An illustration of a cloth having a herringbone weave from Antinoöpolis in Greece from 130 AD. The Falkirk Tartan, a wool 2:2 herringbone tartan found at Vindolanda in England from around 240 AD.
“The pattern was pervasive to ancient life, it even shows up in religious texts and beliefs. It even emerged as a fabric in ancient Ireland. It was a common fabric found throughout the Old World. However, once the Roman Empire fell, the herringbone pattern all but disappeared.”
https://www.padandquill.com/blog/post/t … erringbone