Richard Almack states, [1]
A de Barnardiston, in the reign of Richard I, had two sons, William and Simon. The latter was the father of Walter de Barnardiston, who left two sons, one is called "Alexander de Walpole, son and heir of Walter de Barnardiston," and the other "Roger."
William, son and heir of the first mentioned, was contemporary to King Henry III, and left a son named Geoffrey, who lived in the reign of Edward I, and appears to have married the heiress of the family of Novo Mercato or Newmarch, probably the daughter of Roger de Novo Mercato, but for a considerable time, Amicia, the widow of John de Novo Mercato, his brother, continued to be tenant for life in the Ketton Manor estate and advowson.
[1] Richard Almack, "Keddington alias Ketton, and the Barnardiston Family," Proceedings of the Suffolk Institute of Archaeology 4 (1874), 123-182, at 123, [GoogleBooks].